Decentralised systems
Semitopology, blockchain models and mathematical structures for cooperation without central control.
Research
My work connects mathematical logic and theoretical computer science, from the structure of names and binding to models of decentralised action. Click on a topic to jump to papers on that topic.
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Cryptography
Arithmetisation of first-order logic and cryptographic proofs over finite models.
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Nominal sets and foundations
A mathematical account of names, binding, freshness, and alpha-equivalence, grounded in nominal sets and the Gabbay–Pitts model. The NEW quantifier means
“for a fresh name”
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Logic and proof theory
Nominal algebra, quantifiers, proof search, equational reasoning and the structure of logical derivations.
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Semantics and duality
Mathematical models of logic and computation, including nominal semantics, lattices, topology and representation theorems.
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Rewriting and unification
Rewriting, substitution and unification in the presence of names, binding and alpha-equivalence.
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Lambda calculus and metaprogramming
Lambda-calculi, contexts, staged computation and syntax with meta-variables.
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Programming languages and processes
Programming with binders, types, operational semantics and process calculi.
Selected publications
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Publication type: Journal article
Semitopology: a topological approach to decentralized collaborative action
Journal of Logic and Computation vol. 35, no. 5 article exae050 Oxford University Press online
- Author PDF for Semitopology: a topological approach to decentralized collaborative action
- Publisher / DOI for Semitopology: a topological approach to decentralized collaborative action
- BibTeX for Semitopology: a topological approach to decentralized collaborative action @article{gabbay:semitopology-jlc, author = {Murdoch J. Gabbay}, title = {{Semitopology: a topological approach to decentralized collaborative action}}, journal = {Journal of Logic and Computation}, volume = {35}, number = {5}, eid = {exae050}, publisher = {Oxford University Press}, year = {2025}, month = jul, note = {Article exae050}, doi = {10.1093/logcom/exae050}, eprint = {2303.09287}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1093/logcom/exae050} }
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Publication type: Journal article
Arithmetising Logic: Polynomial Semantics of FOL
Journal of Applied Logics vol. 12, no. 6 pp. 1481–1548 London: College Publications ISSN 2631-9810
- Author PDF for Arithmetising Logic: Polynomial Semantics of FOL
- Journal issue PDF for Arithmetising Logic: Polynomial Semantics of FOL
- BibTeX for Arithmetising Logic: Polynomial Semantics of FOL @article{gabbay:arithmetising-logic, author = {Murdoch J. Gabbay}, title = {{Arithmetising Logic: Polynomial Semantics of FOL}}, journal = {Journal of Applied Logics}, volume = {12}, number = {6}, pages = {1481--1548}, publisher = {College Publications}, address = {London}, year = {2025}, month = oct, issn = {2631-9810}, eprint = {2024/954}, url = {https://www.collegepublications.co.uk/downloads/ifcolog00074.pdf} }
- Repository record for Arithmetising Logic: Polynomial Semantics of FOL
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Publication type: Journal article
Algebras of UTxO blockchains
Mathematical Structures in Computer Science vol. 31, no. 9 In Homage to Martin Hofmann pp. 1034–1089 Cambridge University Press issue dated online
- Author PDF for Algebras of UTxO blockchains
- Publisher / DOI for Algebras of UTxO blockchains
- BibTeX for Algebras of UTxO blockchains @article{gabbay:algub, author = {Murdoch J. Gabbay}, title = {{Algebras of UTxO blockchains}}, journal = {Mathematical Structures in Computer Science}, volume = {31}, number = {9}, pages = {1034--1089}, publisher = {Cambridge University Press}, year = {2022}, month = jan, note = {Special issue: In Homage to Martin Hofmann}, doi = {10.1017/S0960129521000438}, url = {https://doi.org/10.1017/S0960129521000438} }
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Publication type: Journal article
Equivariant ZFA and the foundations of nominal techniques
Journal of Logic and Computation vol. 30, no. 2 pp. 525–548 ISSN 0955-792X online
A new take on the foundations of nominal techniques.
- Author PDF for Equivariant ZFA and the foundations of nominal techniques
- Publisher / DOI for Equivariant ZFA and the foundations of nominal techniques
- BibTeX for Equivariant ZFA and the foundations of nominal techniques @article{gabbay:equzfn, author = {Murdoch J. Gabbay}, title = {{Equivariant ZFA and the foundations of nominal techniques}}, journal = {Journal of Logic and Computation}, volume = {30}, number = {2}, pages = {525--548}, year = {2020}, month = mar, doi = {10.1093/logcom/exz015}, url = {https://doi.org/10.1093/logcom/exz015} }
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Publication type: Journal article
Semantics out of context: nominal absolute denotations for first-order logic and computation
Journal of the ACM vol. 63, no. 3 pp. 1–66 article 25 ISSN 1535-9921
Everything you wanted to know about semantics of first-order logic in nominal sets, topologies, and lattices. A sister paper is Stone duality for first-order logic. Available also on ArXiv.
- Author PDF for Semantics out of context: nominal absolute denotations for first-order logic and computation
- Publisher / DOI for Semantics out of context: nominal absolute denotations for first-order logic and computation
- BibTeX for Semantics out of context: nominal absolute denotations for first-order logic and computation @article{gabbay:semooc, author = {Murdoch J. Gabbay}, title = {{Semantics out of context: nominal absolute denotations for first-order logic and computation}}, journal = {Journal of the ACM}, volume = {63}, number = {3}, eid = {25}, pages = {1--66}, year = {2016}, month = jun, doi = {10.1145/2700819}, url = {https://doi.org/10.1145/2700819} }
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Publication type: Journal article
Foundations of nominal techniques: logic and semantics of variables in abstract syntax
Bulletin of Symbolic Logic vol. 17, no. 2 pp. 161–229
A survey and update paper describing nominal techniques, with specific application to nominal abstract syntax. A sequel to this paper is Nominal terms and nominal logics.
- Author PDF for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- Publisher / DOI for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- BibTeX for Foundations of nominal techniques: logic and semantics of variables in abstract syntax @article{gabbay:fountl, author = {Murdoch J. Gabbay}, title = {{Foundations of nominal techniques: logic and semantics of variables in abstract syntax}}, journal = {Bulletin of Symbolic Logic}, volume = {17}, number = {2}, pages = {161--229}, year = {2011}, month = jun, doi = {10.2178/bsl/1305810911}, url = {https://doi.org/10.2178/bsl/1305810911} }
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Publication type: Journal article
A New Approach to Abstract Syntax with Variable Binding
Formal Aspects of Computing vol. 13, no. 3–5 pp. 341–363
The paper that started it all, at least for me. We introduce nominal sets, atoms-abstraction, the NEW quantifier, and nominal abstract syntax (inductive datatypes of syntax-with-binding) and their inductive reasoning principles. See also the conference version in LICS99.
- Author PDF for A New Approach to Abstract Syntax with Variable Binding
- Publisher / DOI for A New Approach to Abstract Syntax with Variable Binding
- BibTeX for A New Approach to Abstract Syntax with Variable Binding @article{gabbay:newaas-jv, author = {Murdoch J. Gabbay and Andrew M. Pitts}, title = {{A New Approach to Abstract Syntax with Variable Binding}}, journal = {Formal Aspects of Computing}, volume = {13}, number = {3--5}, pages = {341--363}, year = {2002}, month = jul, doi = {10.1007/s001650200016}, url = {https://doi.org/10.1007/s001650200016} }
Full publications list
108 items, arranged by date.
Publications by date
2026
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Publication type: Conference paper
Equational Reasoning in Languages with Binders via Permutation Fixed-Points
11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026) ed. Frank Pfenning Leibniz International Proceedings in Informatics (LIPIcs) vol. 378 pp. 10:1–10:23 article 10 Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik ISBN 978-3-95977-433-8 ISSN 1868-8969
- Author PDF for Equational Reasoning in Languages with Binders via Permutation Fixed-Points
- Publisher / DOI for Equational Reasoning in Languages with Binders via Permutation Fixed-Points
- BibTeX for Equational Reasoning in Languages with Binders via Permutation Fixed-Points @inproceedings{caires-santos-fernandez-gabbay-nantes-sobrinho:permutation-fixed-points, author = {Ali K. Caires-Santos and Maribel Fern{\'a}ndez and Murdoch James Gabbay and Daniele Nantes-Sobrinho}, title = {{Equational Reasoning in Languages with Binders via Permutation Fixed-Points}}, booktitle = {11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)}, editor = {Frank Pfenning}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, volume = {378}, pages = {10:1--10:23}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, year = {2026}, month = jul, isbn = {978-3-95977-433-8}, issn = {1868-8969}, doi = {10.4230/LIPIcs.FSCD.2026.10}, url = {https://doi.org/10.4230/LIPIcs.FSCD.2026.10} }
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Publication type: Technical report
Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
Anoma Research Topics Zenodo
- Author PDF for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- Repository for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- DOI 10.5281/zenodo.21074359 for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- BibTeX for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers @techreport{gabbay-sheff:consistent-histories, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2026}, month = jul, doi = {10.5281/zenodo.21074359}, url = {https://doi.org/10.5281/zenodo.21074359} }
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Publication type: Preprint
Cryptographic certificates of validity for trustworthy AI
- Author PDF for Cryptographic certificates of validity for trustworthy AI
- arXiv for Cryptographic certificates of validity for trustworthy AI
- BibTeX for Cryptographic certificates of validity for trustworthy AI @misc{gabbay:cryptographic-certificates-ai, author = {Murdoch J. Gabbay}, title = {{Cryptographic certificates of validity for trustworthy AI}}, howpublished = {arXiv preprint arXiv:2606.23768}, year = {2026}, month = jun, doi = {10.48550/arXiv.2606.23768}, eprint = {2606.23768}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.48550/arXiv.2606.23768} }
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Publication type: Book chapter
Decentralized Collaborative Action: Cryptoeconomics in Space
Cryptoeconomic Theory eds. Melanie Swan, Soichiro Takagi, Frank Witte Advances in Science and Technology in the Age of AI vol. 1 pp. 53–70 Singapore: World Scientific ISBN 978-1-80061-865-7 eISBN 978-1-80061-867-1
- Author PDF for Decentralized Collaborative Action: Cryptoeconomics in Space
- Publisher / DOI for Decentralized Collaborative Action: Cryptoeconomics in Space
- BibTeX for Decentralized Collaborative Action: Cryptoeconomics in Space @incollection{gabbay:cryptoeconomics-in-space, author = {Murdoch J. Gabbay}, title = {{Decentralized Collaborative Action: Cryptoeconomics in Space}}, booktitle = {Cryptoeconomic Theory}, editor = {Melanie Swan and Soichiro Takagi and Frank Witte}, series = {Advances in Science and Technology in the Age of AI}, volume = {1}, pages = {53--70}, publisher = {World Scientific}, address = {Singapore}, year = {2026}, month = may, isbn = {978-1-80061-865-7}, note = {Electronic ISBN 978-1-80061-867-1}, doi = {10.1142/9781800618664_0004}, eprint = {2504.12493}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.1142/9781800618664_0004} }
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Publication type: Preprint
Nominal techniques as an Agda library
- Author PDF for Nominal techniques as an Agda library
- arXiv for Nominal techniques as an Agda library
- BibTeX for Nominal techniques as an Agda library @misc{gabbay-melkonian:nominal-agda-library, author = {Murdoch J. Gabbay and Orestis Melkonian}, title = {{Nominal techniques as an Agda library}}, howpublished = {arXiv preprint arXiv:2603.03968}, year = {2026}, month = mar, doi = {10.48550/arXiv.2603.03968}, eprint = {2603.03968}, archivePrefix = {arXiv}, primaryClass = {cs.PL}, url = {https://doi.org/10.48550/arXiv.2603.03968} }
- Code for Nominal techniques as an Agda library
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Publication type: Journal article
Semiframes: the algebra of semitopologies and actionable coalitions
Mathematical Structures in Computer Science vol. 36 article e3 Cambridge University Press
- Author PDF for Semiframes: the algebra of semitopologies and actionable coalitions
- Publisher / DOI for Semiframes: the algebra of semitopologies and actionable coalitions
- BibTeX for Semiframes: the algebra of semitopologies and actionable coalitions @article{gabbay:semiframes, author = {Murdoch J. Gabbay}, title = {{Semiframes: the algebra of semitopologies and actionable coalitions}}, journal = {Mathematical Structures in Computer Science}, volume = {36}, eid = {e3}, publisher = {Cambridge University Press}, year = {2026}, month = feb, note = {Article e3}, doi = {10.1017/S096012952510042X}, eprint = {2310.00956}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1017/S096012952510042X} }
2025
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Technical report
Heterogeneous trust in reliable broadcast via modal logic and history structures
Anoma Research Topics Zenodo
- Author PDF for Heterogeneous trust in reliable broadcast via modal logic and history structures
- Repository for Heterogeneous trust in reliable broadcast via modal logic and history structures
- DOI 10.5281/zenodo.17636313 for Heterogeneous trust in reliable broadcast via modal logic and history structures
- BibTeX for Heterogeneous trust in reliable broadcast via modal logic and history structures @techreport{gabbay-sheff:heterogeneous-bracha-broadcast, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Heterogeneous trust in reliable broadcast via modal logic and history structures}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2025}, month = nov, doi = {10.5281/zenodo.17636313}, url = {https://doi.org/10.5281/zenodo.17636313} }
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Publication type: Journal article
Arithmetising Logic: Polynomial Semantics of FOL
Journal of Applied Logics vol. 12, no. 6 pp. 1481–1548 London: College Publications ISSN 2631-9810
- Author PDF for Arithmetising Logic: Polynomial Semantics of FOL
- Journal issue PDF for Arithmetising Logic: Polynomial Semantics of FOL
- BibTeX for Arithmetising Logic: Polynomial Semantics of FOL @article{gabbay:arithmetising-logic, author = {Murdoch J. Gabbay}, title = {{Arithmetising Logic: Polynomial Semantics of FOL}}, journal = {Journal of Applied Logics}, volume = {12}, number = {6}, pages = {1481--1548}, publisher = {College Publications}, address = {London}, year = {2025}, month = oct, issn = {2631-9810}, eprint = {2024/954}, url = {https://www.collegepublications.co.uk/downloads/ifcolog00074.pdf} }
- Repository record for Arithmetising Logic: Polynomial Semantics of FOL
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Publication type: Journal article
Semitopology: a topological approach to decentralized collaborative action
Journal of Logic and Computation vol. 35, no. 5 article exae050 Oxford University Press online
- Author PDF for Semitopology: a topological approach to decentralized collaborative action
- Publisher / DOI for Semitopology: a topological approach to decentralized collaborative action
- BibTeX for Semitopology: a topological approach to decentralized collaborative action @article{gabbay:semitopology-jlc, author = {Murdoch J. Gabbay}, title = {{Semitopology: a topological approach to decentralized collaborative action}}, journal = {Journal of Logic and Computation}, volume = {35}, number = {5}, eid = {exae050}, publisher = {Oxford University Press}, year = {2025}, month = jul, note = {Article exae050}, doi = {10.1093/logcom/exae050}, eprint = {2303.09287}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1093/logcom/exae050} }
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Publication type: Preprint
A declarative approach to specifying Paxos using three-valued modal fixpoint logic
arXiv latest revision
- Author PDF for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- arXiv for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- BibTeX for A declarative approach to specifying Paxos using three-valued modal fixpoint logic @misc{gabbay-zanolini:declarative-distributed-three-valued, author = {Murdoch J. Gabbay and Luca Zanolini}, title = {{A declarative approach to specifying Paxos using three-valued modal fixpoint logic}}, howpublished = {arXiv preprint arXiv:2502.00892}, year = {2025}, month = feb, doi = {10.48550/arXiv.2502.00892}, eprint = {2502.00892}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2502.00892} }
2024
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Publication type: Book
Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
Studies in Logic vol. 108 320 pp. total London: College Publications edition 1 ISBN 978-1-84890-465-1
- Author PDF for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- Publisher page for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- BibTeX for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic @book{gabbay:semitopology-monograph, author = {Murdoch J. Gabbay}, title = {{Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic}}, volume = {108}, publisher = {College Publications}, address = {London}, edition = {1}, year = {2024}, month = aug, isbn = {978-1-84890-465-1}, note = {320 pages}, eprint = {2402.03253}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://www.collegepublications.co.uk/logic/?00056=} }
2022
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Publication type: Journal article
Algebras of UTxO blockchains
Mathematical Structures in Computer Science vol. 31, no. 9 In Homage to Martin Hofmann pp. 1034–1089 Cambridge University Press issue dated online
We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.
- Author PDF for Algebras of UTxO blockchains
- Publisher / DOI for Algebras of UTxO blockchains
- BibTeX for Algebras of UTxO blockchains @article{gabbay:algub, author = {Murdoch J. Gabbay}, title = {{Algebras of UTxO blockchains}}, journal = {Mathematical Structures in Computer Science}, volume = {31}, number = {9}, pages = {1034--1089}, publisher = {Cambridge University Press}, year = {2022}, month = jan, note = {Special issue: In Homage to Martin Hofmann}, doi = {10.1017/S0960129521000438}, url = {https://doi.org/10.1017/S0960129521000438} }
2021
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Publication type: Conference paper
Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
3rd International Workshop on Formal Methods for Blockchains (FMBC 2021) eds. Bruno Bernardo, Diego Marmsoler Open Access Series in Informatics (OASIcs) vol. 95 pp. 2:1–2:14 article 2 Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik ISBN 978-3-95977-209-9 ISSN 2190-6807
- Author PDF for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
- Publisher / DOI for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
- BibTeX for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard @inproceedings{gabbay-jakobsson-sojakova:money-grows-proof-trees, author = {Murdoch J. Gabbay and Arvid Jakobsson and Kristina Sojakova}, title = {{Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard}}, booktitle = {3rd International Workshop on Formal Methods for Blockchains (FMBC 2021)}, editor = {Bruno Bernardo and Diego Marmsoler}, series = {Open Access Series in Informatics (OASIcs)}, volume = {95}, pages = {2:1--2:14}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, year = {2021}, month = nov, isbn = {978-3-95977-209-9}, issn = {2190-6807}, doi = {10.4230/OASIcs.FMBC.2021.2}, url = {https://doi.org/10.4230/OASIcs.FMBC.2021.2} }
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Publication type: Book chapter
Karl Marx and the Blockchain
Media, Technology and Education in a Post-Truth Society ed. Grech, Alex pp. 225–241 Emerald Publishing Limited, Bingley
We consider blockchain technology from a historical, social, and economic perspective, through a lens of the economic theories of Karl Marx.
- Author PDF for Karl Marx and the Blockchain
- Publisher / DOI for Karl Marx and the Blockchain
- BibTeX for Karl Marx and the Blockchain @incollection{gabbay:karmb, author = {Devraj Basu and Murdoch J. Gabbay}, title = {{Karl Marx and the Blockchain}}, booktitle = {Media, Technology and Education in a Post-Truth Society}, pages = {225--241}, publisher = {Emerald Publishing Limited, Bingley}, year = {2021}, month = jul, doi = {10.1108/978-1-80043-906-120211016}, url = {https://doi.org/10.1108/978-1-80043-906-120211016} }
2020
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Publication type: Conference paper
UTxO- vs account-based smart contract blockchain programming paradigms
Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020) eds. Tiziana Margaria, Bernhard Steffen Lecture Notes in Computer Science vol. 12478 pp. 73–88 Springer
We implement two versions of a simple but illustrative smart contract: one in Solidity on the Ethereum blockchain platform, and one in Plutus on the Cardano platform, with annotated code excerpts and with source code attached. We prove some simple but novel results about observational equivalence and conclude with a wide-ranging and detailed discussion in the light of the examples, mathematical model, and mathematical results so far.
- Author PDF for UTxO- vs account-based smart contract blockchain programming paradigms
- Publisher / DOI for UTxO- vs account-based smart contract blockchain programming paradigms
- BibTeX for UTxO- vs account-based smart contract blockchain programming paradigms @inproceedings{gabbay:utxabs, author = {Lars Br{\"u}njes and Murdoch J. Gabbay}, title = {{UTxO- vs account-based smart contract blockchain programming paradigms}}, booktitle = {Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020)}, editor = {Tiziana Margaria and Bernhard Steffen}, series = {Lecture Notes in Computer Science}, volume = {12478}, pages = {73--88}, publisher = {Springer}, year = {2020}, month = oct, doi = {10.1007/978-3-030-61467-6_6}, eprint = {2003.14271}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1007/978-3-030-61467-6_6} }
- arXiv preprint for UTxO- vs account-based smart contract blockchain programming paradigms
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Publication type: Journal article
Equivariant ZFA and the foundations of nominal techniques
Journal of Logic and Computation vol. 30, no. 2 pp. 525–548 ISSN 0955-792X online
A new take on the foundations of nominal techniques.
- Author PDF for Equivariant ZFA and the foundations of nominal techniques
- Publisher / DOI for Equivariant ZFA and the foundations of nominal techniques
- BibTeX for Equivariant ZFA and the foundations of nominal techniques @article{gabbay:equzfn, author = {Murdoch J. Gabbay}, title = {{Equivariant ZFA and the foundations of nominal techniques}}, journal = {Journal of Logic and Computation}, volume = {30}, number = {2}, pages = {525--548}, year = {2020}, month = mar, doi = {10.1093/logcom/exz015}, url = {https://doi.org/10.1093/logcom/exz015} }
2018
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Publication type: Journal article
The language of Stratified Sets is confluent and strongly normalising
Logical Methods in Computer Science vol. 14, no. 2 article 12
Quine’s stratification condition viewed from the perspective of the theory of rewriting.
- Author PDF for The language of Stratified Sets is confluent and strongly normalising
- Publisher / DOI for The language of Stratified Sets is confluent and strongly normalising
- BibTeX for The language of Stratified Sets is confluent and strongly normalising @article{gabbay:lanssc, author = {Murdoch J. Gabbay}, title = {{The language of Stratified Sets is confluent and strongly normalising}}, journal = {Logical Methods in Computer Science}, volume = {14}, number = {2}, eid = {12}, year = {2018}, month = may, note = {Article 12}, doi = {10.23638/LMCS-14(2:12)2018}, url = {https://doi.org/10.23638/LMCS-14(2:12)2018} }
2017
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Publication type: Journal article
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
Annals of Pure and Applied Logic vol. 168, no. 3 pp. 501–621 Elsevier
Everything you wanted to know about semantics of the lambda-calculus using nominal techniques.
- Author PDF for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- Publisher / DOI for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- BibTeX for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness @article{gabbay:repdul, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness}}, journal = {Annals of Pure and Applied Logic}, volume = {168}, number = {3}, pages = {501--621}, publisher = {Elsevier}, year = {2017}, month = mar, doi = {10.1016/j.apal.2016.10.001}, url = {https://doi.org/10.1016/j.apal.2016.10.001} }
2016
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Publication type: Journal article
Checking Overlaps of Nominal Rewriting Rules
Electronic Notes in Theoretical Computer Science vol. 323 pp. 39–56 Elsevier B.V. ISSN 1571-0661
- Repository PDF for Checking Overlaps of Nominal Rewriting Rules
- Publisher / DOI for Checking Overlaps of Nominal Rewriting Rules
- BibTeX for Checking Overlaps of Nominal Rewriting Rules @article{ayalarincon:cheonr, author = {Mauricio Ayala-Rinc{\'o}n and Maribel Fern{\'a}ndez and Murdoch James Gabbay and Ana Cristina Rocha-Oliveira}, title = {{Checking Overlaps of Nominal Rewriting Rules}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {323}, pages = {39--56}, publisher = {Elsevier B.V.}, year = {2016}, month = jul, doi = {10.1016/j.entcs.2016.06.004}, url = {https://doi.org/10.1016/j.entcs.2016.06.004} }
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Publication type: Journal article
Semantics out of context: nominal absolute denotations for first-order logic and computation
Journal of the ACM vol. 63, no. 3 pp. 1–66 article 25 ISSN 1535-9921
Everything you wanted to know about semantics of first-order logic in nominal sets, topologies, and lattices. A sister paper is Stone duality for first-order logic. Available also on ArXiv.
- Author PDF for Semantics out of context: nominal absolute denotations for first-order logic and computation
- Publisher / DOI for Semantics out of context: nominal absolute denotations for first-order logic and computation
- BibTeX for Semantics out of context: nominal absolute denotations for first-order logic and computation @article{gabbay:semooc, author = {Murdoch J. Gabbay}, title = {{Semantics out of context: nominal absolute denotations for first-order logic and computation}}, journal = {Journal of the ACM}, volume = {63}, number = {3}, eid = {25}, pages = {1--66}, year = {2016}, month = jun, doi = {10.1145/2700819}, url = {https://doi.org/10.1145/2700819} }
2015
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Publication type: Conference paper
Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
24th EACSL Annual Conference on Computer Science Logic (CSL 2015) ed. Kreutzer, Stephan Leibniz International Proceedings in Informatics (LIPIcs) vol. 41 pp. 374–389 Dagstuhl, Germany: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik ISBN 978-3-939897-90-3 ISSN 1868-8969
Split “νa.t” as “〈a t a〉”. If that looks simple, try “〈a〈b ab a〉b〉” or “〈a〈b abb〉”.
- Author PDF for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
- Publisher / DOI for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
- BibTeX for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes @inproceedings{gabbay:leatnn, author = {Murdoch J. Gabbay and Dan R. Ghica and Daniela Petrisan}, title = {{Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, volume = {41}, pages = {374--389}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}, year = {2015}, month = sep, doi = {10.4230/LIPIcs.CSL.2015.374}, url = {https://doi.org/10.4230/LIPIcs.CSL.2015.374} }
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Publication type: Journal article
Quantifiers in logic and proof-search using permissive-nominal terms and sets
Journal of Logic and Computation vol. 25, no. 2 pp. 473–523
Nominal terms used to model the γ- and δ-rules controlling quantifiers in reductive proof search.
- Author PDF for Quantifiers in logic and proof-search using permissive-nominal terms and sets
- Publisher / DOI for Quantifiers in logic and proof-search using permissive-nominal terms and sets
- BibTeX for Quantifiers in logic and proof-search using permissive-nominal terms and sets @article{gabbay:qualps, author = {Murdoch J. Gabbay and Claus-Peter Wirth}, title = {{Quantifiers in logic and proof-search using permissive-nominal terms and sets}}, journal = {Journal of Logic and Computation}, volume = {25}, number = {2}, pages = {473--523}, year = {2015}, month = apr, doi = {10.1093/logcom/exs057}, url = {https://doi.org/10.1093/logcom/exs057} }
2014
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Publication type: Journal article
A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
The IfCoLog Journal of Logics and their Applications vol. 1, no. 2 pp. 83–106 College Publications ISBN 978-1-84890-163-6 ISSN 2055-3706
- Repository PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- Journal issue PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- DOI 10.17863/CAM.42150 for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- BibTeX for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality @article{gabbay:simcmt, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality}}, journal = {The IfCoLog Journal of Logics and their Applications}, volume = {1}, number = {2}, pages = {83--106}, publisher = {College Publications}, year = {2014}, month = nov, isbn = {978-1-84890-163-6}, issn = {2055-3706}, doi = {10.17863/CAM.42150}, url = {https://doi.org/10.17863/CAM.42150} }
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Publication type: Book chapter
Stone duality for First-Order Logic: a nominal approach
HOWARD-60. A Festschrift on the Occasion of Howard Barringer's 60th Birthday eds. Andrei Voronkov, Margarita Korovina EPiC Series in Computing vol. 42 pp. 178–209 EasyChair
Stone duality for first-order logic. Open predicates map to open sets. No valuations. See also the ancestor of this paper, Stone duality for nominal Boolean algebras with NEW. This version of the paper differs from the version that appeared in the Festschrift in the correction of some small typos and errors.
- Author PDF for Stone duality for First-Order Logic: a nominal approach
- Publisher / DOI for Stone duality for First-Order Logic: a nominal approach
- BibTeX for Stone duality for First-Order Logic: a nominal approach @incollection{gabbay:stodfo, author = {Murdoch J. Gabbay}, title = {{Stone duality for First-Order Logic: a nominal approach}}, booktitle = {HOWARD-60. A Festschrift on the Occasion of Howard Barringer's 60th Birthday}, editor = {Andrei Voronkov and Margarita Korovina}, series = {EPiC Series in Computing}, volume = {42}, pages = {178--209}, publisher = {EasyChair}, year = {2014}, month = feb, doi = {10.29007/tp3z}, url = {https://doi.org/10.29007/tp3z} }
2013
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Publication type: Journal article
Imaginary groups: lazy monoids and reversible computation.
Mathematical Structures in Computer Science vol. 23, no. 05 pp. 1002–1031 Cambridge University Press
A duality between partially ordered groups and an (apparently novel) category of partially ordered monoids and lazy homomorphisms between them. Every monoid can be lazily injected into a group. Reading a monoid as an abstract notion of computation (or at least, of composition), this construction shows how to add reversibility.
- Author PDF for Imaginary groups: lazy monoids and reversible computation.
- Publisher / DOI for Imaginary groups: lazy monoids and reversible computation.
- BibTeX for Imaginary groups: lazy monoids and reversible computation. @article{gabbay:imaglm, author = {Murdoch J. Gabbay and Peter Kropholler}, title = {{Imaginary groups: lazy monoids and reversible computation.}}, journal = {Mathematical Structures in Computer Science}, volume = {23}, number = {05}, pages = {1002--1031}, publisher = {Cambridge University Press}, year = {2013}, month = oct, doi = {10.1017/S0960129512000849}, url = {https://doi.org/10.1017/S0960129512000849} }
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Publication type: Journal article
Denotation of contextual modal type theory (CMTT): syntax and meta-programming
Journal of Applied Logic vol. 11, no. 1 pp. 1–29
A denotation for CMTT. I wanted to understand this for years, and now we have an answer!
- Author PDF for Denotation of contextual modal type theory (CMTT): syntax and meta-programming
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- BibTeX for Denotation of contextual modal type theory (CMTT): syntax and meta-programming @article{gabbay:dencmt, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of contextual modal type theory (CMTT): syntax and meta-programming}}, journal = {Journal of Applied Logic}, volume = {11}, number = {1}, pages = {1--29}, year = {2013}, month = mar, doi = {10.1016/j.jal.2012.07.002}, url = {https://doi.org/10.1016/j.jal.2012.07.002} }
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Publication type: Book chapter
Nominal terms and nominal logics: from foundations to meta-mathematics
Handbook of Philosophical Logic vol. 17 pp. 79–178 Springer
A survey and update chapter describing a journey from (permissive-)nominal sets, via nominal terms, rewriting and algebra, to permissive-nominal logic and a finite first-order nominal axiomatisation of arithmetic. See also Foundations of nominal techniques.
- Author PDF for Nominal terms and nominal logics: from foundations to meta-mathematics
- Publisher / DOI for Nominal terms and nominal logics: from foundations to meta-mathematics
- BibTeX for Nominal terms and nominal logics: from foundations to meta-mathematics @incollection{gabbay:nomtnl, author = {Murdoch J. Gabbay}, title = {{Nominal terms and nominal logics: from foundations to meta-mathematics}}, booktitle = {Handbook of Philosophical Logic}, volume = {17}, pages = {79--178}, publisher = {Springer}, year = {2013}, month = jan, doi = {10.1007/978-94-007-6600-6_2}, url = {https://doi.org/10.1007/978-94-007-6600-6_2} }
2012
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Publication type: Journal article
Meta-variables as infinite lists in nominal terms unification and rewriting
Logic Journal of the IGPL vol. 20, no. 6 pp. 967–1000 Oxford University Press
Nominal terms unknowns modelled concretely as infinite lists of distinct atoms. Theorems simplify and many even become immediate corollaries of equivariance.
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- Publisher / DOI for Meta-variables as infinite lists in nominal terms unification and rewriting
- BibTeX for Meta-variables as infinite lists in nominal terms unification and rewriting @article{gabbay:metvil, author = {Murdoch J. Gabbay}, title = {{Meta-variables as infinite lists in nominal terms unification and rewriting}}, journal = {Logic Journal of the IGPL}, volume = {20}, number = {6}, pages = {967--1000}, publisher = {Oxford University Press}, year = {2012}, month = dec, doi = {10.1093/jigpal/jzs005}, url = {https://doi.org/10.1093/jigpal/jzs005} }
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Publication type: Journal article
Nominal SOS
Electronic Notes in Theoretical Computer Science vol. 286 Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012) pp. 103–116 Elsevier ISSN 1571-0661
- Publisher / DOI for Nominal SOS
- BibTeX for Nominal SOS @article{cimini-mousavi-reniers-gabbay:nomsos, author = {Matteo Cimini and Mohammad Reza Mousavi and Michel A. Reniers and Murdoch J. Gabbay}, title = {{Nominal SOS}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {286}, pages = {103--116}, publisher = {Elsevier}, year = {2012}, month = sep, issn = {1571-0661}, note = {Special issue: Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012)}, doi = {10.1016/j.entcs.2012.08.008}, url = {https://doi.org/10.1016/j.entcs.2012.08.008} }
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Publication type: Journal article
Game semantics in the nominal model
Electronic Notes in Theoretical Computer Science vol. 286 Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012) pp. 173–189 Elsevier
Nominal semantics used to model and generalise pointer sequences, using a nice new idea of nominal coabstraction. The result: an expression of game semantics using nominal-style atoms.
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- BibTeX for Game semantics in the nominal model @article{gabbay:gamsnm, author = {Murdoch J. Gabbay and Dan R. Ghica}, title = {{Game semantics in the nominal model}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {286}, pages = {173--189}, publisher = {Elsevier}, year = {2012}, month = sep, note = {Special issue: Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012)}, doi = {10.1016/j.entcs.2012.08.012}, url = {https://doi.org/10.1016/j.entcs.2012.08.012} }
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Publication type: Journal article
PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
Theoretical Computer Science vol. 451 pp. 38–69
We translate PNL to HOL (permissive-nominal logic to higher-order logic). Notably, this involved turning a nominal atoms-abstraction into a total function.
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- Publisher / DOI for PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
- BibTeX for PNL to HOL: from the logic of nominal sets to the logic of higher-order functions @article{gabbay:pnlthf, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{PNL to HOL: from the logic of nominal sets to the logic of higher-order functions}}, journal = {Theoretical Computer Science}, volume = {451}, pages = {38--69}, year = {2012}, month = sep, doi = {10.1016/j.tcs.2012.06.007}, url = {https://doi.org/10.1016/j.tcs.2012.06.007} }
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Publication type: Journal article
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
Journal of Symbolic Logic vol. 77, no. 3 pp. 828–852 Duke University Press
Moving between permissive-nominal (infinitely-supported) and nominal (finitely-supported) models, with applications to permissive-nominal algebra completeness and permissive-nominal logic completeness. Cool stuff playing with nominal sets.
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- BibTeX for Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free @article{gabbay:finisn, author = {Murdoch J. Gabbay}, title = {{Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free}}, journal = {Journal of Symbolic Logic}, volume = {77}, number = {3}, pages = {828--852}, publisher = {Duke University Press}, year = {2012}, month = sep, doi = {10.2178/jsl/1344862164}, url = {https://doi.org/10.2178/jsl/1344862164} }
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Publication type: Journal article
Permissive-nominal logic: First-order logic over nominal terms and sets
ACM Transactions on Computational Logic vol. 13, no. 3 pp. 1–36 article 20 New York: Association for Computing Machinery
First-order logic with term-formers that can bind names. So we can axiomatise things like forall or lambda, or finitely axiomatise arithmetic. This gives PNL much of the expressive power of higher-order logic: but terms, derivations and models of PNL are first-order and the logic seems to strike a good balance between expressivity and simplicity. In brief, PNL is obtained by adding universal quantification to nominal algebra. But, to make that work we needed to invent permission sets.
This is a journal version of the conference paper.
- Author PDF for Permissive-nominal logic: First-order logic over nominal terms and sets
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- BibTeX for Permissive-nominal logic: First-order logic over nominal terms and sets @article{gabbay:pernl-jv, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Permissive-nominal logic: First-order logic over nominal terms and sets}}, journal = {ACM Transactions on Computational Logic}, volume = {13}, number = {3}, eid = {20}, pages = {1--36}, publisher = {Association for Computing Machinery}, year = {2012}, month = aug, doi = {10.1145/2287718.2287720}, url = {https://doi.org/10.1145/2287718.2287720} }
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Publication type: Journal article
Unity in nominal equational reasoning: The algebra of equality on nominal sets
Journal of Applied Logic vol. 10, no. 2 pp. 199–217
An exhaustive analysis of the design choices involved in nominal equational / equality / algebra logic over nominal sets.
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- BibTeX for Unity in nominal equational reasoning: The algebra of equality on nominal sets @article{gabbay:uniner, author = {Murdoch J. Gabbay}, title = {{Unity in nominal equational reasoning: The algebra of equality on nominal sets}}, journal = {Journal of Applied Logic}, volume = {10}, number = {2}, pages = {199--217}, year = {2012}, month = jun, doi = {10.1016/j.jal.2012.03.001}, url = {https://doi.org/10.1016/j.jal.2012.03.001} }
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Publication type: Conference paper
Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
Proceedings of the 9th Italian Convention on Computational Logic (CILC 2012) ed. Francesca A. Lisi CEUR workshop proceedings vol. 857 pp. 104–118 CEUR-WS.org
Universal lattice properties used to specify top, meet, and forall as instances of a single notion of fresh-finite limit.
- Author PDF for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
- Proceedings record for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
- BibTeX for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits @inproceedings{gabbay:nomspl, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits}}, booktitle = {Proceedings of the 9th Italian Convention on Computational Logic (CILC 2012)}, editor = {Francesca A. Lisi}, series = {CEUR workshop proceedings}, volume = {857}, pages = {104--118}, publisher = {CEUR-WS.org}, year = {2012}, month = may, url = {https://nbn-resolving.de/urn:nbn:de:0074-857-8} }
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Publication type: Preprint
Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
Subsumed by “Denotation of contextual modal type theory (CMTT): syntax and meta-programming”.
- Author PDF for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- arXiv record (1202.0904) for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- BibTeX for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT) @misc{gabbay:densmc, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)}}, howpublished = {arXiv preprint arXiv:1202.0904}, year = {2012}, month = feb, eprint = {1202.0904}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/1202.0904} }
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Publication type: Workshop abstract
Quantifier rules in reductive proof using nominal semantics
Automated Reasoning Workshop (ARW 2012) 2 pp. total
We consider liberalised delta-rules in proof-search, and give them a nominal algebraic semantics.
- Author PDF for Quantifier rules in reductive proof using nominal semantics
- BibTeX for Quantifier rules in reductive proof using nominal semantics @inproceedings{gabbay:quarrp-ea, author = {Murdoch J. Gabbay}, title = {{Quantifier rules in reductive proof using nominal semantics}}, booktitle = {Automated Reasoning Workshop (ARW 2012)}, year = {2012}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/quarrp-ea.pdf} }
- Catalogue record for Quantifier rules in reductive proof using nominal semantics
2011
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Publication type: Preprint
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Subsumed by “PNL to HOL: from the logic of nominal sets to the logic of higher-order functions”.
- Author PDF for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
- arXiv record (1111.4611) for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
- BibTeX for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions @misc{gabbay:nomsbf-arxiv, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions}}, howpublished = {arXiv preprint arXiv:1111.4611}, year = {2011}, month = nov, eprint = {1111.4611}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/1111.4611} }
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Publication type: Journal article
Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
Mathematical Structures in Computer Science vol. 21, no. 5 pp. 997–1033 Cambridge University Press
A nominal denotational semantics for meta-variables. Nominal techniques have already given semantics to variables; in this paper we show how they can do the same for meta-variables. The version on this webpage includes corrections.
- Author PDF for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- Publisher / DOI for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- BibTeX for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables @article{gabbay:twolns, author = {Murdoch J. Gabbay}, title = {{Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables}}, journal = {Mathematical Structures in Computer Science}, volume = {21}, number = {5}, pages = {997--1033}, publisher = {Cambridge University Press}, year = {2011}, month = oct, doi = {10.1017/S0960129511000272}, url = {https://doi.org/10.1017/S0960129511000272} }
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Publication type: Conference paper
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011) EPTCS vol. 71 pp. 58–75
Henkin-style models of the lambda-calculus in nominal sets. No Tarski-style valuations; instead we use nominal substitution algebras. The title of the paper in the pre-proceedings was Semantics for functions of the simply-typed lambda-calculus with nominal sets.
- Author PDF for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- Publisher / DOI for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- BibTeX for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets @inproceedings{gabbay:nomhss, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets}}, booktitle = {Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011)}, series = {EPTCS}, volume = {71}, pages = {58--75}, year = {2011}, month = sep, doi = {10.4204/EPTCS.71.5}, url = {https://doi.org/10.4204/EPTCS.71.5} }
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Publication type: Conference paper
Principal Types for Nominal Theories
Proceedings of the 18th International Symposium on Fundamentals of Computation Theory (FCT 2011) eds. Olaf Owe, Martin Steffen, Jan Arne Telle Lecture Notes in Computer Science vol. 6914 pp. 160–172 Berlin, Heidelberg: Springer ISBN 978-3-642-22952-7 eISBN 978-3-642-22953-4
Rank 1 polymorphic types for nominal rewriting and algebra. So you want to type rules like beta- or eta-conversion over nominal terms? This paper describes one way to do it.
- Author PDF for Principal Types for Nominal Theories
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- BibTeX for Principal Types for Nominal Theories @inproceedings{gabbay:pritnt, author = {Elliot Fairweather and Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Principal Types for Nominal Theories}}, booktitle = {Proceedings of the 18th International Symposium on Fundamentals of Computation Theory (FCT 2011)}, editor = {Olaf Owe and Martin Steffen and Jan Arne Telle}, series = {Lecture Notes in Computer Science}, volume = {6914}, pages = {160--172}, publisher = {Springer}, address = {Berlin, Heidelberg}, year = {2011}, month = aug, isbn = {978-3-642-22952-7}, note = {Electronic ISBN 978-3-642-22953-4}, doi = {10.1007/978-3-642-22953-4_14}, url = {https://doi.org/10.1007/978-3-642-22953-4_14} }
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Publication type: Conference paper
Stone duality for nominal Boolean algebras with NEW
Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011) Lecture Notes in Computer Science vol. 6859 pp. 192–207 Springer ISBN 978-3-642-22943-5
The NEW quantifier is the canonical name-generation/name-restriction operator for predicates on the nominal model of names and binding. We give a sets representation theorem and Stone-type duality. See also the subsequent Stone duality for first-order logic.
- Author PDF for Stone duality for nominal Boolean algebras with NEW
- Publisher / DOI for Stone duality for nominal Boolean algebras with NEW
- BibTeX for Stone duality for nominal Boolean algebras with NEW @inproceedings{gabbay:stodnb, author = {Murdoch J. Gabbay and Tadeusz Litak and Daniela Petri{\c{s}}an}, title = {{Stone duality for nominal Boolean algebras with NEW}}, booktitle = {Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011)}, series = {Lecture Notes in Computer Science}, volume = {6859}, pages = {192--207}, publisher = {Springer}, year = {2011}, month = aug, doi = {10.1007/978-3-642-22944-2_14}, url = {https://doi.org/10.1007/978-3-642-22944-2_14} }
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Publication type: Journal article
Foundations of nominal techniques: logic and semantics of variables in abstract syntax
Bulletin of Symbolic Logic vol. 17, no. 2 pp. 161–229
A survey and update paper describing nominal techniques, with specific application to nominal abstract syntax. A sequel to this paper is Nominal terms and nominal logics.
- Author PDF for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- Publisher / DOI for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- BibTeX for Foundations of nominal techniques: logic and semantics of variables in abstract syntax @article{gabbay:fountl, author = {Murdoch J. Gabbay}, title = {{Foundations of nominal techniques: logic and semantics of variables in abstract syntax}}, journal = {Bulletin of Symbolic Logic}, volume = {17}, number = {2}, pages = {161--229}, year = {2011}, month = jun, doi = {10.2178/bsl/1305810911}, url = {https://doi.org/10.2178/bsl/1305810911} }
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Publication type: Conference paper
Freshness and name-restriction in sets of traces with names
Foundations of software science and computation structures, 14th International Conference (FOSSACS 2011) Lecture Notes in Computer Science vol. 6604 pp. 365–380 Springer
A concrete sets model of name-generation/name-restriction using a relative of Gabbay-Pitts atoms-abstraction. The technical innovation is the notion of a positive subset of a nominal set.
- Author PDF for Freshness and name-restriction in sets of traces with names
- Publisher / DOI for Freshness and name-restriction in sets of traces with names
- BibTeX for Freshness and name-restriction in sets of traces with names @inproceedings{gabbay:frenrs, author = {Murdoch J. Gabbay and Vincenzo Ciancia}, title = {{Freshness and name-restriction in sets of traces with names}}, booktitle = {Foundations of software science and computation structures, 14th International Conference (FOSSACS 2011)}, series = {Lecture Notes in Computer Science}, volume = {6604}, pages = {365--380}, publisher = {Springer}, year = {2011}, month = mar, doi = {10.1007/978-3-642-19805-2_25}, url = {https://doi.org/10.1007/978-3-642-19805-2_25} }
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Publication type: Workshop abstract
Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
Automated Reasoning Workshop (ARW 2011) 2 pp. total
We sketch what theorem-proving might look like if it were based on nominal terms. This eventually evolved into Nominal terms and nominal logics.
- Author PDF for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
- BibTeX for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover @inproceedings{gabbay:metbnt-ea, author = {Murdoch J. Gabbay}, title = {{Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover}}, booktitle = {Automated Reasoning Workshop (ARW 2011)}, year = {2011}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/metbnt-ea.pdf} }
- Catalogue record for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
2010
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Publication type: Conference paper
A simple class of Kripke-style models in which logic and computation have equal standing
International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010) eds. Edmund M. Clarke, Andrei Voronkov Lecture Notes in Computer Science vol. 6355 pp. 231–254 Springer
A sound and complete model of lambda-calculus reductions in structures related to Kripke structures. We construct a logic for the models and identify lambda-terms with predicates by direct compositional translation. Reduction becomes logical entailment.
- Author PDF for A simple class of Kripke-style models in which logic and computation have equal standing
- Publisher / DOI for A simple class of Kripke-style models in which logic and computation have equal standing
- BibTeX for A simple class of Kripke-style models in which logic and computation have equal standing @inproceedings{gabbay:simcks, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A simple class of Kripke-style models in which logic and computation have equal standing}}, booktitle = {International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010)}, editor = {Edmund M. Clarke and Andrei Voronkov}, series = {Lecture Notes in Computer Science}, volume = {6355}, pages = {231--254}, publisher = {Springer}, year = {2010}, month = dec, doi = {10.1007/978-3-642-17511-4_14}, url = {https://doi.org/10.1007/978-3-642-17511-4_14} }
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Publication type: Journal article
Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
Logic Journal of the IGPL vol. 18, no. 6 pp. 769–822
You cannot quotient nominal terms by alpha-equivalence, but you can quotient permissive nominal terms. This journal paper extends the conference version and technical report and includes some elements from another conference paper. My version includes minor corrections to the published version.
- Author PDF for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
- Publisher / DOI for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
- BibTeX for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version) @article{gabbay:perntu-jv, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)}}, journal = {Logic Journal of the IGPL}, volume = {18}, number = {6}, pages = {769--822}, year = {2010}, month = dec, doi = {10.1093/jigpal/jzq006}, url = {https://doi.org/10.1093/jigpal/jzq006} }
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Publication type: Conference paper
Closed nominal rewriting and efficiently computable nominal algebra equality
Electronic Proceedings in Theoretical Computer Science vol. 34 pp. 37–51
We relate nominal algebra and rewriting, with concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality, which includes specifications of lambda-calculus and first-order logic.
- Author PDF for Closed nominal rewriting and efficiently computable nominal algebra equality
- Publisher / DOI for Closed nominal rewriting and efficiently computable nominal algebra equality
- BibTeX for Closed nominal rewriting and efficiently computable nominal algebra equality @inproceedings{gabbay:clonre, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Closed nominal rewriting and efficiently computable nominal algebra equality}}, booktitle = {Electronic Proceedings in Theoretical Computer Science}, volume = {34}, pages = {37--51}, year = {2010}, month = sep, doi = {10.4204/EPTCS.34.5}, url = {https://doi.org/10.4204/EPTCS.34.5} }
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Publication type: Conference paper
Permissive Nominal Logic
Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010) pp. 165–176 New York: ACM Press
First-order logic with binding term-formers. We can axiomatise forall or lambda, and finitely axiomatise arithmetic. PNL has much of the expressive power of higher-order logic but terms, derivations and models of PNL are first-order. PNL is nominal algebra” + quantification over nominal unknowns. See also the journal version.
- Author PDF for Permissive Nominal Logic
- Publisher / DOI for Permissive Nominal Logic
- BibTeX for Permissive Nominal Logic @inproceedings{gabbay:pernl, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Permissive Nominal Logic}}, booktitle = {Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010)}, pages = {165--176}, publisher = {ACM Press}, year = {2010}, month = jul, doi = {10.1145/1836089.1836111}, url = {https://doi.org/10.1145/1836089.1836111} }
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Publication type: Technical report
An observation on support and freshness in nominal sets
School of Mathematical and Computer Sciences, Heriot-Watt University Technical Report no. HW-MACS-TR-0077 14 pp. total Edinburgh, UK
There are two natural ways of excluding an atom a in nominal techniques: we can consider the sets X such that a is fresh for X, or we can consider the sets X such that a is fresh for every x ∈ X. We show these lead to isomorphic categories. This can be read as saying that a fresh atom in the object-level is categorically isomorphic to a fresh atom in the meta-level.
- Author PDF for An observation on support and freshness in nominal sets
- BibTeX for An observation on support and freshness in nominal sets @techreport{gabbay:obssfn-tr, author = {Murdoch J. Gabbay}, title = {{An observation on support and freshness in nominal sets}}, institution = {School of Mathematical and Computer Sciences, Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0077}, address = {Edinburgh, UK}, year = {2010}, month = may, note = {14 pages}, url = {https://www.gabbay.org.uk/papers/obssfn-tr.pdf} }
- Catalogue record for An observation on support and freshness in nominal sets
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Publication type: Journal article
A nominal axiomatisation of the lambda-calculus
Journal of Logic and Computation vol. 20, no. 2 pp. 501–531
We can write axioms for the lambda-calculus in nominal algebra—but do we know those axioms are right? We prove completeness of axioms for alpha-beta-(eta) equivalence. This subsumes a previous book chapter.
- Author PDF for A nominal axiomatisation of the lambda-calculus
- Publisher / DOI for A nominal axiomatisation of the lambda-calculus
- BibTeX for A nominal axiomatisation of the lambda-calculus @article{gabbay:nomalc, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{A nominal axiomatisation of the lambda-calculus}}, journal = {Journal of Logic and Computation}, volume = {20}, number = {2}, pages = {501--531}, year = {2010}, month = apr, doi = {10.1093/logcom/exp049}, url = {https://doi.org/10.1093/logcom/exp049} }
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Publication type: Journal article
Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
Information and Computation vol. 208, no. 3 pp. 230–258
Types-as-propositions, incomplete (nominal) terms as incomplete derivations. That is, the “variables” of nominal terms become “holes” in first-order logic derivations. A freshness condition becomes a promise that an assumption will be discharged.
This journal paper expands on the conference version.
- Author PDF for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- Publisher / DOI for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- BibTeX for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms @article{gabbay:curhif-jv, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms}}, journal = {Information and Computation}, volume = {208}, number = {3}, pages = {230--258}, year = {2010}, month = mar, doi = {10.1016/j.ic.2009.09.003}, url = {https://doi.org/10.1016/j.ic.2009.09.003} }
2009
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Publication type: Journal article
Nominal universal algebra: equational logic with names and binding
Journal of Logic and Computation vol. 19, no. 6 pp. 1455–1508
Nominal Algebra, a logic for equational reasoning over nominal sets, which uses nominal terms. Think: “algebra with term-formers that can bind”. This paper subsumes conference versions in WOLLIC07 and NWPT06, and a 2006 technical report, and is followed by Nominal terms, nominal logics.
Many of the papers applying nominal algebra were in print before the journal paper. You might be interested in
- Author PDF for Nominal universal algebra: equational logic with names and binding
- Publisher / DOI for Nominal universal algebra: equational logic with names and binding
- BibTeX for Nominal universal algebra: equational logic with names and binding @article{gabbay:nomuae, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal universal algebra: equational logic with names and binding}}, journal = {Journal of Logic and Computation}, volume = {19}, number = {6}, pages = {1455--1508}, year = {2009}, month = dec, doi = {10.1093/logcom/exp033}, url = {https://doi.org/10.1093/logcom/exp033} }
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Publication type: Journal article
The lambda-context calculus (extended version)
Information and computation vol. 207, no. 12 pp. 1369–1400 Elsevier
The lambda-context calculus is a multi-level lambda-calculus, generalising a two-level lambda-calculus with an infinite hierarchy of variables such that higher level variables behave like meta-variables with respect to lower level variables. It can express both capture-avoiding and capturing substitution (instantiation), explicit substitutions, references, and more.
An extended version of our LFMTP’07 paper.
- Author PDF for The lambda-context calculus (extended version)
- Publisher / DOI for The lambda-context calculus (extended version)
- BibTeX for The lambda-context calculus (extended version) @article{gabbay:lamcce, author = {Murdoch J. Gabbay and St{\'e}phane Lengrand}, title = {{The lambda-context calculus (extended version)}}, journal = {Information and computation}, volume = {207}, number = {12}, pages = {1369--1400}, publisher = {Elsevier}, year = {2009}, month = dec, doi = {10.1016/j.ic.2009.06.004}, url = {https://doi.org/10.1016/j.ic.2009.06.004} }
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Publication type: Conference paper
Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
Proceedings of the 4th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2009) pp. 64–73 ACM
We translate between equality derivations in lambda-theories and equality derivations in permissive-nominal algebra theories. This connects reasoning over higher-order syntax to reasoning over nominal terms.
- Author PDF for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
- Publisher / DOI for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
- BibTeX for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables @inproceedings{gabbay:unialt, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables}}, booktitle = {Proceedings of the 4th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2009)}, pages = {64--73}, publisher = {ACM}, year = {2009}, month = aug, doi = {10.1145/1577824.1577835}, url = {https://doi.org/10.1145/1577824.1577835} }
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Publication type: Conference paper
Two-level lambda-calculus
Electronic Notes in Theoretical Computer Science vol. 246 pp. 107–129
The two-level lambda-calculus gives a functional operational semantics to nominal terms unknowns; the result is a lambda-calculus with capturing and capture-avoiding substitution are represented and nominal terms style alpha-equivalence for level 1 variables (atoms) in the presence of level 2 variables (unknowns).
This ENTCS paper subsumes a 15 page workshop paper entitled “Two-and-a-halfth order lambda-calculus”, which appeared in the preproceedings of WFLP08.
- Author PDF for Two-level lambda-calculus
- Publisher / DOI for Two-level lambda-calculus
- BibTeX for Two-level lambda-calculus @inproceedings{gabbay:twollc, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Two-level lambda-calculus}}, booktitle = {Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008)}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {246}, pages = {107--129}, year = {2009}, month = aug, doi = {10.1016/j.entcs.2009.07.018}, url = {https://doi.org/10.1016/j.entcs.2009.07.018} }
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Publication type: Conference paper
Term sequent logic
Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008) Electronic Notes in Theoretical Computer Science vol. 246 pp. 87–106
Term sequents are sequents for terms instead of predicates. We build lambda-calculus reductions, similar to rewriting but with structure corresponding to proof-theory. This subsumes a 15 page workshop paper of the same title which appeared in the preproceedings of WFLP08.
- Author PDF for Term sequent logic
- Publisher / DOI for Term sequent logic
- BibTeX for Term sequent logic @inproceedings{gabbay:tersl, author = {Michael Gabbay and Murdoch J. Gabbay}, title = {{Term sequent logic}}, booktitle = {Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008)}, series = {Electronic Notes in Theoretical Computer Science}, volume = {246}, pages = {87--106}, year = {2009}, month = aug, doi = {10.1016/j.entcs.2009.07.017}, url = {https://doi.org/10.1016/j.entcs.2009.07.017} }
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Publication type: Conference paper
Permissive Nominal Terms and their Unification
Atti del Convegno CILC 09: 24-esimo Convegno Italiano di Logica Computazionale eds. Marco Gavanelli, Fabrizio Riguzzi pp. 70–84 Ferrara, Italy: GULP (Italian Association for Logic Programming)
Permissive nominal terms resemble nominal terms, but using infinite and co-infinite sets of atoms. We get ‘always fresh’ and ‘always alpha-rename’ properties familiar from first-and higher- order syntax but absent from nominal terms. See also the journal version
- Author PDF for Permissive Nominal Terms and their Unification
- Proceedings PDF for Permissive Nominal Terms and their Unification
- BibTeX for Permissive Nominal Terms and their Unification @inproceedings{gabbay:perntu, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification}}, booktitle = {Atti del Convegno CILC 09: 24-esimo Convegno Italiano di Logica Computazionale}, editor = {Marco Gavanelli and Fabrizio Riguzzi}, pages = {70--84}, publisher = {GULP (Italian Association for Logic Programming)}, address = {Ferrara, Italy}, year = {2009}, month = jun, url = {https://www.programmazionelogica.it/wp-content/uploads/2014/10/cilc09-proceedings.pdf} }
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Publication type: Journal article
Nominal Algebra and the HSP Theorem
Journal of Logic and Computation vol. 19, no. 2 pp. 341–367
It is not obvious that Nominal Algebra is a pure equational logic, because of freshness conditions. E.g. η-conversion (eta-conversion) looks like this: a#X⊢λa.(Xa)=X. This is a conditional equality. The HSPA theorem for Nominal Algebra asserts that any model of a Nominal Algebra theory is part of a quotient of a product of atoms-abstractions of free models.
This limits the complexity of models, and puts a precise measure on the notion of equality of nominal algebra: making things equal, putting things into tuples, and forming atoms-abstracton.
- Author PDF for Nominal Algebra and the HSP Theorem
- Publisher / DOI for Nominal Algebra and the HSP Theorem
- BibTeX for Nominal Algebra and the HSP Theorem @article{gabbay:nomahs, author = {Murdoch J. Gabbay}, title = {{Nominal Algebra and the HSP Theorem}}, journal = {Journal of Logic and Computation}, volume = {19}, number = {2}, pages = {341--367}, year = {2009}, month = apr, doi = {10.1093/logcom/exn055}, url = {https://doi.org/10.1093/logcom/exn055} }
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Publication type: Conference paper
Semantic nominal terms
Second International Workshop on Theory and Applications of Abstraction, Substitution, and Naming (TAASN 2009), associated with ETAPS 2009 5 pp. total York, UK
We reconcile nominal atoms-abstraction with nominal terms. See the journal paper on two-level nominal sets.
- Author PDF for Semantic nominal terms
- BibTeX for Semantic nominal terms @inproceedings{gabbay:semnt, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Semantic nominal terms}}, booktitle = {Second International Workshop on Theory and Applications of Abstraction, Substitution, and Naming (TAASN 2009), associated with ETAPS 2009}, address = {York, UK}, year = {2009}, month = mar, note = {5 pages}, url = {https://www.gabbay.org.uk/papers/semnt-ea.pdf} }
- Later version DOI for Semantic nominal terms
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Publication type: Journal article
A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
Theoretical Computer Science vol. 410, no. 12-13 pp. 1159–1189 Elsevier
Nominal sets contain names and alpha-equivalence, so what about defining a capture-avoiding substitution action for them? We give a notion of “replace a by y in x” where x and y are any two sets.
This subsumes the AISB08 paper, and is followed by Freshness and name-restriction in sets of traces with names.
- Author PDF for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- Publisher / DOI for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- BibTeX for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets @article{gabbay:stusun, author = {Murdoch J. Gabbay}, title = {{A study of substitution, using nominal techniques and Fraenkel-Mostowski sets}}, journal = {Theoretical Computer Science}, volume = {410}, number = {12-13}, pages = {1159--1189}, publisher = {Elsevier}, year = {2009}, month = mar, doi = {10.1016/j.tcs.2008.11.013}, url = {https://doi.org/10.1016/j.tcs.2008.11.013} }
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Publication type: Technical report
Permissive Nominal Terms and their Unification
School of Mathematical and Computer Sciences, Heriot-Watt University Technical Report no. HW-MACS-TR-0062 49 pp. total Edinburgh, UK
We consider permissive nominal terms, and their unification. Permissive nominal terms closely resemble nominal terms, but they recover these useful ‘always fresh’ and ‘always alpha-rename’ properties, familiar from first-and higher- order syntax. In the permissive world, freshness contexts are elided, and the notion of unifier is based on substitution alone, rather than on nominal terms’ notion of substitution plus freshness conditions.
We prove that expressivity is not lost moving to the permissive case and translate from nominal terms unification into permissive nominal terms unification.
- Author PDF for Permissive Nominal Terms and their Unification
- BibTeX for Permissive Nominal Terms and their Unification @techreport{gabbay:perntu-tr, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification}}, institution = {School of Mathematical and Computer Sciences, Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0062}, address = {Edinburgh, UK}, year = {2009}, month = feb, note = {49 pages}, url = {https://www.gabbay.org.uk/papers/perntu-tr.pdf} }
- Later version DOI for Permissive Nominal Terms and their Unification
2008
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Publication type: Book chapter
The lambda-calculus is nominal algebraic
Reasoning in Simple Type Theory: Festschrift in Honor of Peter B. Andrews on his 70th Birthday eds. Christoph Benzmüller, Chad E. Brown, Jörg Siekmann, Richard Statman Studies in Logic vol. 17 pp. 271–302 London: College Publications ISBN 978-1-904987-70-3
We use nominal algebra to give a sound and complete axiomatisation of the lambda-calculus. Subsumed by a journal version.
- Author PDF for The lambda-calculus is nominal algebraic
- Publisher page for The lambda-calculus is nominal algebraic
- BibTeX for The lambda-calculus is nominal algebraic @incollection{gabbay:lamcna, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{The lambda-calculus is nominal algebraic}}, booktitle = {Reasoning in Simple Type Theory: Festschrift in Honor of Peter B. Andrews on his 70th Birthday}, editor = {Christoph Benzm{\"u}ller and Chad E. Brown and J{\"o}rg Siekmann and Richard Statman}, series = {Studies in Logic}, volume = {17}, pages = {271--302}, publisher = {College Publications}, address = {London}, year = {2008}, month = dec, isbn = {978-1-904987-70-3}, url = {https://www.collegepublications.co.uk/logic/mlf/?00010=} }
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Publication type: Conference paper
Nominal renaming sets
Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008) pp. 158–173 Springer
Nominal renaming sets are like nominal sets, but they are based on a finitely-supported renaming action — a renaming is a function from atoms to atoms, not necessarily injective. This provides a convenient sets-based semantics with which to justify Higher-Order Abstract Syntax.
- Author PDF for Nominal renaming sets
- Publisher / DOI for Nominal renaming sets
- BibTeX for Nominal renaming sets @inproceedings{gabbay:nomrs, author = {Murdoch J. Gabbay and Martin Hofmann}, title = {{Nominal renaming sets}}, booktitle = {Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008)}, pages = {158--173}, publisher = {Springer}, year = {2008}, month = nov, doi = {10.1007/978-3-540-89439-1_11}, url = {https://doi.org/10.1007/978-3-540-89439-1_11} }
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Publication type: Technical report
Permissive nominal terms
INRIA Research Report no. RR-6682 15 pp. total Orsay, France
We consider permissive nominal terms, and their unification. Permissive nominal terms closely resemble nominal terms, but they recover these useful ‘always fresh’ and ‘always alpha-rename’ properties, familiar from first-and higher- order syntax. In the permissive world, freshness contexts are elided, and the notion of unifier is based on substitution alone, rather than on nominal terms’ notion of substitution plus freshness conditions.
- Author PDF for Permissive nominal terms
- HAL record for Permissive nominal terms
- BibTeX for Permissive nominal terms @techreport{gabbay:pernt-tr, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive nominal terms}}, institution = {INRIA}, series = {Research Report}, number = {RR-6682}, address = {Orsay, France}, year = {2008}, month = oct, note = {15 pages}, url = {https://inria.hal.science/inria-00335115} }
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Publication type: Journal article
One-and-a-halfth-order Logic
Journal of Logic and Computation vol. 18, no. 4 pp. 521–562
A logic of schematic reasoning based on nominal algebra. For instance, we can represent the schematic assertion “if a does not occur in φ then φ implies ∀a.φ” and its derivation as “a#X⊢X⊃∀a.X” and corresponding schematic derivations. We give algebraic and sequent characterisations, and show correctness with respect to first-order logic. This subsumes the PPDP06 conference paper.
- Author PDF for One-and-a-halfth-order Logic
- Publisher / DOI for One-and-a-halfth-order Logic
- BibTeX for One-and-a-halfth-order Logic @article{gabbay:oneaah-jv, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{One-and-a-halfth-order Logic}}, journal = {Journal of Logic and Computation}, volume = {18}, number = {4}, pages = {521--562}, year = {2008}, month = aug, doi = {10.1093/logcom/exm064}, url = {https://doi.org/10.1093/logcom/exm064} }
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Publication type: Journal article
a-logic with arrows (journal version)
Electronic Notes in Theoretical Computer Science vol. 216 pp. 3–29
a-logic has a predicate ‘atom’ which is true of a term when it is a variable symbol We prove cut-elimination, give a notion of model, prove soundess and completeness, and axiomatise substitution and the lambda-calculus, using ‘atom’ to identify variables. This is a journal version of our WFLP07 paper, with an improved proof-system.
- Author PDF for a-logic with arrows (journal version)
- Publisher / DOI for a-logic with arrows (journal version)
- BibTeX for a-logic with arrows (journal version) @article{gabbay:alwa-jv, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic with arrows (journal version)}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {216}, pages = {3--29}, year = {2008}, month = jul, doi = {10.1016/j.entcs.2008.06.031}, url = {https://doi.org/10.1016/j.entcs.2008.06.031} }
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Publication type: Conference paper
One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008) Lecture Notes in Artificial Intelligence vol. 5110 pp. 179–193 Springer
Nominal terms give Curry-Howard for schematic derivations. Atoms correspond with assumptions, unknowns correspond with schematic parts of the derivation, freshness conditions corresponds with the condition that an assumption will be discharged (even if the schema does not specify how).
- Author PDF for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- Publisher / DOI for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- BibTeX for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations @inproceedings{gabbay:curhid, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations}}, booktitle = {Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008)}, series = {Lecture Notes in Artificial Intelligence}, volume = {5110}, pages = {179--193}, publisher = {Springer}, year = {2008}, month = jun, doi = {10.1007/978-3-540-69937-8_16}, url = {https://doi.org/10.1007/978-3-540-69937-8_16} }
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Publication type: Journal article
Capture-Avoiding Substitution as a Nominal Algebra
Formal Aspects of Computing vol. 20, no. 4-5 pp. 451–479 Springer
Nominal algebra makes it easy to write axioms for capture-avoiding substitution—or does it? Consider the choices: do you want b[a↦X]=b or a#Z⊢Z[a↦X]=Z? Do you want Z[a↦a]=Z or not? In this paper we write down various plausible axioms and prove senses in which they are correct.
This is a journal version of our ICTAC06 paper.
- Author PDF for Capture-Avoiding Substitution as a Nominal Algebra
- Publisher / DOI for Capture-Avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-Avoiding Substitution as a Nominal Algebra @article{gabbay:capasn-jv, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-Avoiding Substitution as a Nominal Algebra}}, journal = {Formal Aspects of Computing}, volume = {20}, number = {4-5}, pages = {451--479}, publisher = {Springer}, year = {2008}, month = jun, doi = {10.1007/s00165-007-0056-1}, url = {https://doi.org/10.1007/s00165-007-0056-1} }
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Publication type: Conference paper
Substitution for Fraenkel-Mostowski foundations
Proceedings of the 2008 AISB Symposium on Computing and Philosophy AISB 2008 Convention: Communication, Interaction and Social Intelligence vol. 12 pp. 65–72 University of Aberdeen, Aberdeen, UK: The Society for the Study of Artificial Intelligence and Simulation of Behaviour ISBN 1-902956-71-0
We present a capture-avoiding substitution action defined in Fraenkel-Mostowski set theory on the entire sets universe. Substitution is commonly understood as an action on syntax (for example when building a logic or a lambda calculus) and thus as a prelude to, but not a part of, mathematical denotations. We show that it also has denotational realisation, as an operation on elements of a mathematical foundation.
- Author PDF for Substitution for Fraenkel-Mostowski foundations
- Proceedings PDF for Substitution for Fraenkel-Mostowski foundations
- BibTeX for Substitution for Fraenkel-Mostowski foundations @inproceedings{gabbay:subfmf, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Substitution for Fraenkel-Mostowski foundations}}, booktitle = {Proceedings of the 2008 AISB Symposium on Computing and Philosophy}, series = {AISB 2008 Convention: Communication, Interaction and Social Intelligence}, volume = {12}, pages = {65--72}, publisher = {The Society for the Study of Artificial Intelligence and Simulation of Behaviour}, address = {University of Aberdeen, Aberdeen, UK}, year = {2008}, month = apr, isbn = {1-902956-71-0}, url = {https://aisb.org.uk/wp-content/uploads/2019/12/Final-vol-12.pdf} }
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Publication type: Workshop abstract
Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics 2 pp. total Paris, France
- Author PDF for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
- BibTeX for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants @inproceedings{gabbay:towpta, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants}}, booktitle = {Paris Arch{\'e} workshop on abstract objects in semantics and the philosophy of mathematics}, address = {Paris, France}, year = {2008}, month = feb, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/towpta.pdf} }
- Catalogue record for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
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Publication type: Workshop abstract
Arbitrary Objects in Mathematics and Semantics
Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics 2 pp. total Paris, France
A photo of Michael delivering the talk.
- Author PDF for Arbitrary Objects in Mathematics and Semantics
- BibTeX for Arbitrary Objects in Mathematics and Semantics @inproceedings{gabbay:arboms, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Arbitrary Objects in Mathematics and Semantics}}, booktitle = {Paris Arch{\'e} workshop on abstract objects in semantics and the philosophy of mathematics}, address = {Paris, France}, year = {2008}, month = feb, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/arboms.pdf} }
- Catalogue record for Arbitrary Objects in Mathematics and Semantics
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Publication type: Journal article
The lambda-context calculus
Electronic Notes in Theoretical Computer Science vol. 196 pp. 19–35 Elsevier ISSN 1571-0661
We present a simple lambda-calculus whose syntax is populated by variables which behave like meta-variables. It can express both capture-avoiding and capturing substitution (instantiation).
- Author PDF for The lambda-context calculus
- Publisher / DOI for The lambda-context calculus
- BibTeX for The lambda-context calculus @article{gabbay:lamcc, author = {Murdoch J. Gabbay and St{\'e}phane Lengrand}, title = {{The lambda-context calculus}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {196}, pages = {19--35}, publisher = {Elsevier}, year = {2008}, month = jan, doi = {10.1016/j.entcs.2007.09.015}, url = {https://doi.org/10.1016/j.entcs.2007.09.015} }
2007
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Publication type: Technical report
Nominal renaming sets
Heriot-Watt University Technical Report no. HW-MACS-TR-0058 35 pp. total Edinburgh, UK
Subsumed by my LPAR08 paper with Martin Hofmann. Nominal sets based on a renaming action (possibly non-injective functions on atoms) instead of the more usual permutation action.
- Author PDF for Nominal renaming sets
- BibTeX for Nominal renaming sets @techreport{gabbay:nomrs-tr, author = {Murdoch J. Gabbay}, title = {{Nominal renaming sets}}, institution = {Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0058}, address = {Edinburgh, UK}, year = {2007}, month = dec, note = {35 pages}, url = {https://www.gabbay.org.uk/papers/nomrs-tr.pdf} }
- Later version DOI for Nominal renaming sets
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Publication type: Technical report
Nominal Algebra and the HSP Theorem
Heriot-Watt University HW-MACS-TR-0057
Subsumed by “Nominal Algebra and the HSP Theorem”.
We prove the HSP (Homomorphism, Subalgebra, Product) theorem for Nominal Algebra, also known as Birkhoffs theorem.
- Author PDF for Nominal Algebra and the HSP Theorem
- Institutional record for Nominal Algebra and the HSP Theorem
- BibTeX for Nominal Algebra and the HSP Theorem @techreport{gabbay:nomahs-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra and the HSP Theorem}}, institution = {Heriot-Watt University}, number = {HW-MACS-TR-0057}, year = {2007}, month = nov, url = {https://research.tue.nl/en/publications/nominal-algebra-and-the-hsp-theorem/} }
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Publication type: Book chapter
Curry-style types for nominal terms
Types for Proofs and Programs (TYPES 06) Lecture Notes in Computer Science vol. 4502 pp. 125–139 Springer
Nominal terms are a representation of syntax-with-binding in the presence of explicit meta-variables and alpha-renaming. Making this consistent with a functional type system is non-trivial because the same meta-variable may appear in different contexts and under different binders, so when that meta-variable is instantiated the same variable name may be injected into different contexts of abstractions.
- Author PDF for Curry-style types for nominal terms
- Publisher / DOI for Curry-style types for nominal terms
- BibTeX for Curry-style types for nominal terms @incollection{gabbay:curstn, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Curry-style types for nominal terms}}, booktitle = {Types for Proofs and Programs (TYPES 06)}, series = {Lecture Notes in Computer Science}, volume = {4502}, pages = {125--139}, publisher = {Springer}, year = {2007}, month = sep, doi = {10.1007/978-3-540-74464-1_9}, url = {https://doi.org/10.1007/978-3-540-74464-1_9} }
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Publication type: Conference paper
A Formal Calculus for Informal Equality with Binding
WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation Lecture Notes in Computer Science vol. 4576 pp. 162–176 Springer
Nominal Algebra, a logic for equality reasoning with freshness and binding, based on nominal terms. See also applications Capture-avoiding substitution as a nominal algebra and One-and-a-halfth order logic in 2006, and the later journal version.
- Author PDF for A Formal Calculus for Informal Equality with Binding
- Publisher / DOI for A Formal Calculus for Informal Equality with Binding
- BibTeX for A Formal Calculus for Informal Equality with Binding @inproceedings{gabbay:forcie, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{A Formal Calculus for Informal Equality with Binding}}, booktitle = {WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation}, series = {Lecture Notes in Computer Science}, volume = {4576}, pages = {162--176}, publisher = {Springer}, year = {2007}, month = jul, doi = {10.1007/978-3-540-73445-1_12}, url = {https://doi.org/10.1007/978-3-540-73445-1_12} }
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Publication type: Journal article
A General Mathematics of Names
Information and Computation vol. 205, no. 7 pp. 982–1011 Academic Press
Nominal sets as introduced here have an atoms-abstraction operation generalising alpha-equivalence. In this paper we consider infinite simultaneous alpha-equivalence. The trick is to interpret small as well-orderable.
- Author PDF for A General Mathematics of Names
- Publisher / DOI for A General Mathematics of Names
- BibTeX for A General Mathematics of Names @article{gabbay:genmn, author = {Murdoch J. Gabbay}, title = {{A General Mathematics of Names}}, journal = {Information and Computation}, volume = {205}, number = {7}, pages = {982--1011}, publisher = {Academic Press}, year = {2007}, month = jul, doi = {10.1016/j.ic.2006.10.010}, url = {https://doi.org/10.1016/j.ic.2006.10.010} }
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Publication type: Conference paper
a-logic with arrows
Proceedings of the 16th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2007), held at RDP 2007 pp. 47–62 Paris, France
Subsumed by the journal version. A previous paper on a-logic includes a predicate ‘atom’ related to ‘isvar&rsquo from PROLOG; to identify a term as a variable symbol. We give an overhauled proof-system, and a new denotational semantics, which use a directed notion of equality similar to a rewrite relation. We get improved ability to axiomatise logic and computation. See also the proceedings of the WFLP07 workshop.
- Author PDF for a-logic with arrows
- BibTeX for a-logic with arrows @inproceedings{gabbay:alwa, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic with arrows}}, booktitle = {Proceedings of the 16th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2007), held at RDP 2007}, pages = {47--62}, address = {Paris, France}, year = {2007}, month = jun, url = {https://www.gabbay.org.uk/papers/alwa.pdf} }
- Later version DOI for a-logic with arrows
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Publication type: Journal article
Nominal rewriting (journal version)
Information and Computation vol. 205, no. 6 pp. 917–965
Rewriting on nominal terms. An example: a#X⊢λa.(Xa)→X (eta-contraction). Nominal techniques changed from being about syntax-with-binding to being about specifications-with-binding. We also consider criteria for confluence. See the PPDP04 conference paper and Nominal universal algebra and Closed nominal rewriting and efficiently computable nominal algebra equality.
- Author PDF for Nominal rewriting (journal version)
- Publisher / DOI for Nominal rewriting (journal version)
- BibTeX for Nominal rewriting (journal version) @article{gabbay:nomr-jv, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Nominal rewriting (journal version)}}, journal = {Information and Computation}, volume = {205}, number = {6}, pages = {917--965}, year = {2007}, month = jun, doi = {10.1016/j.ic.2006.12.002}, url = {https://doi.org/10.1016/j.ic.2006.12.002} }
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Publication type: Journal article
Hierarchical Nominal Terms and Their Theory of Rewriting
Electronic Notes in Theoretical Computer Science vol. 174, no. 5 pp. 37–52 Elsevier ISSN 1571-0661
So nominal terms have two levels of variable. But that just means we have internalised the meta-level once. What if we do this repeatedly until we reach a fixedpoint? We build a theory of rewriting on nominal terms with an infinite hierarchy of variables. We examine case studies, including capture- and non-capture avoiding substitution and the NEW calculus of contexts as rewrite systems.
This is a revised and expanded version of my LFMTP06 workshop paper.
- Author PDF for Hierarchical Nominal Terms and Their Theory of Rewriting
- Publisher / DOI for Hierarchical Nominal Terms and Their Theory of Rewriting
- BibTeX for Hierarchical Nominal Terms and Their Theory of Rewriting @article{gabbay:hienr, author = {Murdoch J. Gabbay}, title = {{Hierarchical Nominal Terms and Their Theory of Rewriting}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {174}, number = {5}, pages = {37--52}, publisher = {Elsevier}, year = {2007}, month = jun, doi = {10.1016/j.entcs.2007.01.017}, url = {https://doi.org/10.1016/j.entcs.2007.01.017} }
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Publication type: Journal article
Fresh Logic: proof-theory and semantics for FM and nominal techniques
Journal of Applied Logic vol. 5, no. 2 pp. 356–387 Elsevier
The first proof-theory for the NEW quantifier, with sound and complete semantics. See also the subsequent LICS paper. The conference paper came into print three years earlier, though it was written later.
- Author PDF for Fresh Logic: proof-theory and semantics for FM and nominal techniques
- Publisher / DOI for Fresh Logic: proof-theory and semantics for FM and nominal techniques
- BibTeX for Fresh Logic: proof-theory and semantics for FM and nominal techniques @article{gabbay:frelog, author = {Murdoch J. Gabbay}, title = {{Fresh Logic: proof-theory and semantics for FM and nominal techniques}}, journal = {Journal of Applied Logic}, volume = {5}, number = {2}, pages = {356--387}, publisher = {Elsevier}, year = {2007}, month = jun, doi = {10.1016/j.jal.2005.10.012}, url = {https://doi.org/10.1016/j.jal.2005.10.012} }
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Publication type: Technical report
Capture-Avoiding Substitution as a Nominal Algebra
Heriot-Watt University HW-MACS-TR-0053
Subsumed by “Capture-Avoiding Substitution as a Nominal Algebra”.
Decidability and theory of models of a nominal algebra axiomatisation of capture-avoiding substitution.
- Author PDF for Capture-Avoiding Substitution as a Nominal Algebra
- Institutional record for Capture-Avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-Avoiding Substitution as a Nominal Algebra @techreport{gabbay:capasn-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-Avoiding Substitution as a Nominal Algebra}}, institution = {Heriot-Watt University}, number = {HW-MACS-TR-0053}, year = {2007}, url = {https://research.tue.nl/en/publications/capture-avoiding-substitution-as-a-nominal-algebra-3/} }
2006
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Publication type: Conference paper
Capture-avoiding Substitution as a Nominal Algebra
Proceedings of the 3rd International Colloquium on Theoretical Aspects of Computing (ICTAC 2006) Lecture Notes in Computer Science vol. 4281 pp. 198–212 Berlin: Springer
We give a detailed investigation of the fundamental operation which is capture-avoiding substitution, investigated from the point of view of Nominal Algebra. See also the journal version.
- Author PDF for Capture-avoiding Substitution as a Nominal Algebra
- Publisher / DOI for Capture-avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-avoiding Substitution as a Nominal Algebra @inproceedings{gabbay:capasn, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-avoiding Substitution as a Nominal Algebra}}, booktitle = {Proceedings of the 3rd International Colloquium on Theoretical Aspects of Computing (ICTAC 2006)}, series = {Lecture Notes in Computer Science}, volume = {4281}, pages = {198--212}, publisher = {Springer}, year = {2006}, month = nov, doi = {10.1007/11921240_14}, url = {https://doi.org/10.1007/11921240_14} }
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Publication type: Workshop abstract
Nominal SOS
Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06) 2 pp. total Reykjavík, Iceland
We describe Nominal Structured Operational Semantics, a framework for specifying operational semantics within which proofs of congruence are carried out ‘once and for all’.
- Author PDF for Nominal SOS
- Workshop PDF for Nominal SOS
- BibTeX for Nominal SOS @inproceedings{gabbay:nomsos-nwpt, author = {MohammadReza Mousavi and Michel A. Reniers and Murdoch J. Gabbay}, title = {{Nominal SOS}}, booktitle = {Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06)}, address = {Reykjav{\'i}k, Iceland}, year = {2006}, month = oct, note = {2 pages}, url = {https://staff.ru.is/luca/NWPT06/PAPERS/paper1.pdf} }
- Later version DOI for Nominal SOS
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Publication type: Workshop abstract
Nominal Algebra
Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06) 3 pp. total Reykjavík, Iceland
We describe Nominal Algebra, an algebraic system which can directly encode systems-with-binding without using functions. See also Slides of the accompanying talk, and a paper.
- Author PDF for Nominal Algebra
- Institutional record for Nominal Algebra
- BibTeX for Nominal Algebra @inproceedings{gabbay:noma-nwpt, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra}}, booktitle = {Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06)}, address = {Reykjav{\'i}k, Iceland}, year = {2006}, month = oct, note = {3 pages}, url = {https://research.tue.nl/en/publications/nominal-algebra/} }
- Later version DOI for Nominal Algebra
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Publication type: Conference paper
Hierarchical nominal terms and their theory of rewriting
Proceedings of LFMTP’06: First International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice eds. Alberto Momigliano, Brigitte Pientka pp. 32–45 Seattle, Washington, USA
- Proceedings PDF for Hierarchical nominal terms and their theory of rewriting
- BibTeX for Hierarchical nominal terms and their theory of rewriting @inproceedings{gabbay:hienr-preproceedings, author = {Murdoch J. Gabbay}, title = {{Hierarchical nominal terms and their theory of rewriting}}, booktitle = {Proceedings of LFMTP'06: First International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice}, editor = {Alberto Momigliano and Brigitte Pientka}, pages = {32--45}, address = {Seattle, Washington, USA}, year = {2006}, month = aug, url = {https://www.cs.mcgill.ca/~bpientka/lfmtp.pdf} }
- Later version DOI for Hierarchical nominal terms and their theory of rewriting
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Publication type: Conference paper
One-and-a-halfth-order logic
Proceedings of the 8th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2006) pp. 189–200 ACM ISBN 1-59593-388-3
A logic for schematic first-order reasoning; unknown predicates are nominal unknowns in the sense of nominal terms and may occur under binders. We characterise derivability using nominal algebra and using sequents, prove equivalence, cut-elimination, and correctness. See the journal version.
- Author PDF for One-and-a-halfth-order logic
- Publisher / DOI for One-and-a-halfth-order logic
- BibTeX for One-and-a-halfth-order logic @inproceedings{gabbay:oneaah, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{One-and-a-halfth-order logic}}, booktitle = {Proceedings of the 8th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2006)}, pages = {189--200}, publisher = {ACM}, year = {2006}, month = jul, doi = {10.1145/1140335.1140359}, url = {https://doi.org/10.1145/1140335.1140359} }
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Publication type: Technical report
Nominal Algebra
Heriot-Watt University Technical Report no. HW-MACS-TR-0045 13 pp. total Edinburgh, UK
A detailed treatment of the nominal algebra presented in our abstract in NWPT06. Subsumed by a journal version.
- Author PDF for Nominal Algebra
- Institutional record for Nominal Algebra
- BibTeX for Nominal Algebra @techreport{gabbay:noma-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra}}, institution = {Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0045}, address = {Edinburgh, UK}, year = {2006}, note = {13 pages}, url = {https://research.tue.nl/en/publications/nominal-algebra-2/} }
- Later version DOI for Nominal Algebra
2005
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Publication type: Book chapter
Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
We Will Show Them: Essays in Honour of Dov Gabbay eds. S. Artemov, H. Barringer, A. d'Avila Garcez, Luis C. Lamb, J. Woods vol. 1 pp. 701–730 London: College Publications ISBN 1-904987-11-7 (pb); 1-904987-25-7 (hb)
We investigate Restart a natural deduction rule which reads “from A we may deduce B”. This powerful rule by Dov Gabbay in the 1980s adds many theorems to intuitionistic logic; we investigate them and consider generalisations. This is not a hoax.
- Author PDF for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- Publisher page for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- BibTeX for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning @incollection{gabbay:somfcg, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning}}, booktitle = {We Will Show Them: Essays in Honour of Dov Gabbay}, editor = {S. Artemov and H. Barringer and A. d'Avila Garcez and Luis C. Lamb and J. Woods}, volume = {1}, pages = {701--730}, publisher = {College Publications}, address = {London}, year = {2005}, month = oct, isbn = {1-904987-11-7 (pb); 1-904987-25-7 (hb)}, url = {https://www.collegepublications.co.uk/tributes/?00002} }
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Publication type: Book chapter
a-logic
We Will Show Them: Essays in Honour of Dov Gabbay eds. S. Artemov, H. Barringer, A. d'Avila Garcez, Luis C. Lamb, J. Woods vol. 1 pp. 731–764 London: College Publications ISBN 1-904987-11-7 (pb); 1-904987-25-7 (hb)
We extend first-order logic with a predicate expressing the intensional property of a term “this is a variable symbol”. We axiomatise substition, object-level binding, and thus exhibit a finite first-order axiomatisation of the lambda-calculus. See also a related development.
- Author PDF for a-logic
- Publisher page for a-logic
- BibTeX for a-logic @incollection{gabbay:alogic, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic}}, booktitle = {We Will Show Them: Essays in Honour of Dov Gabbay}, editor = {S. Artemov and H. Barringer and A. d'Avila Garcez and Luis C. Lamb and J. Woods}, volume = {1}, pages = {731--764}, publisher = {College Publications}, address = {London}, year = {2005}, month = oct, isbn = {1-904987-11-7 (pb); 1-904987-25-7 (hb)}, url = {https://www.collegepublications.co.uk/tributes/?00002} }
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Publication type: Conference paper
SOS for Higher Order Processes (Extended Abstract)
Proceedings of the 16th International Conference on Concurrency Theory (CONCUR'05) eds. Abadi, Martin; de Alfaro, Luca Lecture Notes in Computer Science vol. 3653 pp. 308–322 Springer
Promoted PANTH is a Structural Operational Semantics framework extends tyft/tyxt, for which a notion of higher-order bisimulation is a congruence. (Slides by MohammadReza Mousavi.)
- Author PDF for SOS for Higher Order Processes (Extended Abstract)
- Publisher / DOI for SOS for Higher Order Processes (Extended Abstract)
- BibTeX for SOS for Higher Order Processes (Extended Abstract) @inproceedings{gabbay:soshop, author = {MohammadReza Mousavi and Murdoch J. Gabbay and Michel A. Reniers}, title = {{SOS for Higher Order Processes (Extended Abstract)}}, booktitle = {Proceedings of the 16th International Conference on Concurrency Theory (CONCUR'05)}, series = {Lecture Notes in Computer Science}, volume = {3653}, pages = {308--322}, publisher = {Springer}, year = {2005}, month = sep, doi = {10.1007/11539452_25}, url = {https://doi.org/10.1007/11539452_25} }
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Publication type: Conference paper
A NEW calculus of contexts
Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05) eds. Pedro Barahona, Amy P. Felty pp. 94–105 Lisbon, Portugal: Association for Computing Machinery ISBN 1-59593-090-6
Subsumed by a later journal paper. A lambda-calculus with context substitution (which does not avoid capture) alongside ordinary beta-reduction. It can encode dynamic binding, objects, and interesting properties associated to partial evaluation. Meta-properties include an applicative characterisation of contextual equivalence.
- Author PDF for A NEW calculus of contexts
- Publisher / DOI for A NEW calculus of contexts
- BibTeX for A NEW calculus of contexts @inproceedings{gabbay:newcc, author = {Murdoch J. Gabbay}, title = {{A NEW calculus of contexts}}, booktitle = {Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05)}, editor = {Pedro Barahona and Amy P. Felty}, pages = {94--105}, publisher = {Association for Computing Machinery}, address = {Lisbon, Portugal}, year = {2005}, month = jul, isbn = {1-59593-090-6}, doi = {10.1145/1069774.1069783}, url = {https://doi.org/10.1145/1069774.1069783} }
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Publication type: Conference paper
Nominal rewriting with name generation: abstraction vs. locality
Proceedings of the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2005) pp. 47–58 ACM Press
We generalise and extend previous work on rewriting with variable binding, with name generation/restriction, and study its properties.
- Author PDF for Nominal rewriting with name generation: abstraction vs. locality
- Publisher / DOI for Nominal rewriting with name generation: abstraction vs. locality
- BibTeX for Nominal rewriting with name generation: abstraction vs. locality @inproceedings{gabbay:nomrng, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Nominal rewriting with name generation: abstraction vs. locality}}, booktitle = {Proceedings of the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2005)}, pages = {47--58}, publisher = {ACM Press}, year = {2005}, month = jul, doi = {10.1145/1069774.1069779}, url = {https://doi.org/10.1145/1069774.1069779} }
2004
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Publication type: Journal article
Nominal Unification
Theoretical Computer Science vol. 323, no. 1–3 pp. 473–497 Essex, UK: Elsevier ISSN 0304-3975
This paper introduced nominal terms, which extend first-order terms with term-formers that can bind. Unification of nominal terms is decidable and has most general unifiers. This journal paper subsumes a conference version. The next step up in abstraction from nominal terms is their permissive variant.
- Author PDF for Nominal Unification
- Publisher / DOI for Nominal Unification
- BibTeX for Nominal Unification @article{gabbay:nomu-jv, author = {Christian Urban and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{Nominal Unification}}, journal = {Theoretical Computer Science}, volume = {323}, number = {1--3}, pages = {473--497}, publisher = {Elsevier}, year = {2004}, month = sep, doi = {10.1016/j.tcs.2004.06.016}, url = {https://doi.org/10.1016/j.tcs.2004.06.016} }
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Publication type: Conference paper
Nominal Rewriting Systems
Proceedings of the 6th ACM SIGPLAN symposium on Principles and Practice of Declarative Programming (PPDP 2004) pp. 108–119 ACM Press
We generalise first-order rewriting with terms involving binding operations, using a nominal approach in which bound entities are explicitly named.
- Author PDF for Nominal Rewriting Systems
- Publisher / DOI for Nominal Rewriting Systems
- BibTeX for Nominal Rewriting Systems @inproceedings{gabbay:nomr, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay and Ian Mackie}, title = {{Nominal Rewriting Systems}}, booktitle = {Proceedings of the 6th ACM SIGPLAN symposium on Principles and Practice of Declarative Programming (PPDP 2004)}, pages = {108--119}, publisher = {ACM Press}, year = {2004}, month = aug, doi = {10.1145/1013963.1013978}, url = {https://doi.org/10.1145/1013963.1013978} }
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Publication type: Conference paper
A Sequent Calculus for Nominal Logic
Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004) pp. 139–148 IEEE Computer Society
A sequent calculus for Nominal Logic. We consider notions of Logic Programming in it and give a translation of Miller and Tiu’s FOLN.
- Author PDF for A Sequent Calculus for Nominal Logic
- Publisher / DOI for A Sequent Calculus for Nominal Logic
- BibTeX for A Sequent Calculus for Nominal Logic @inproceedings{gabbay:seqcnl, author = {Murdoch J. Gabbay and James Cheney}, title = {{A Sequent Calculus for Nominal Logic}}, booktitle = {Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004)}, pages = {139--148}, publisher = {IEEE Computer Society}, year = {2004}, month = jul, doi = {10.1109/LICS.2004.6}, url = {https://doi.org/10.1109/LICS.2004.6} }
2003
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Publication type: Conference paper
Nominal Unification
Proceedings of the 17th International Workshop on Computer Science Logic (CSL 2003) Lecture Notes in Computer Science vol. 2803 pp. 513–527 Springer
See here for formal verification of the results Urban in Isabelle.
- Author PDF for Nominal Unification
- Publisher / DOI for Nominal Unification
- BibTeX for Nominal Unification @inproceedings{gabbay:nomu, author = {Christian Urban and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{Nominal Unification}}, booktitle = {Proceedings of the 17th International Workshop on Computer Science Logic (CSL 2003)}, series = {Lecture Notes in Computer Science}, volume = {2803}, pages = {513--527}, publisher = {Springer}, year = {2003}, month = dec, doi = {10.1007/978-3-540-45220-1_41}, url = {https://doi.org/10.1007/978-3-540-45220-1_41} }
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Publication type: Book chapter
The π-Calculus in FM
Thirty-five years of Automating Mathematics ed. Fairouz D. Kamareddine Applied Logic Series vol. 28 pp. 247–269 Dordrecht: Kluwer Academic Publishers edition 1 ISBN 978-1-4020-1656-1 eISBN 978-94-017-0253-9
We discuss how to use nominal techniques to model name-generation, specifically in a process calculus, though the techniques use are probably more generally applicable. (Electronic versions differ in minor corrections from the printed version.)
- Author PDF for The π-Calculus in FM
- Publisher / DOI for The π-Calculus in FM
- BibTeX for The π-Calculus in FM @incollection{gabbay:picfm, author = {Murdoch J. Gabbay}, title = {{The {$\pi$}-Calculus in FM}}, booktitle = {Thirty-five years of Automating Mathematics}, editor = {Fairouz D. Kamareddine}, series = {Applied Logic Series}, volume = {28}, pages = {247--269}, publisher = {Kluwer Academic Publishers}, address = {Dordrecht}, edition = {1}, year = {2003}, month = nov, isbn = {978-1-4020-1656-1}, note = {Electronic ISBN 978-94-017-0253-9}, doi = {10.1007/978-94-017-0253-9_10}, url = {https://doi.org/10.1007/978-94-017-0253-9_10} }
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Publication type: Conference paper
FreshML: Programming with Binders Made Simple
SIGPLAN Notices vol. 38 pp. 263–274 ACM Press
ML enriched with Gabbay-Pitts atoms-abstraction and nominal name-generation and pattern-matching.
- Author PDF for FreshML: Programming with Binders Made Simple
- Publisher / DOI for FreshML: Programming with Binders Made Simple
- BibTeX for FreshML: Programming with Binders Made Simple @inproceedings{gabbay:frepbm, author = {Mark R. Shinwell and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{FreshML: Programming with Binders Made Simple}}, booktitle = {Proceedings of the 8th ACM SIGPLAN International Conference on Functional Programming (ICFP 2003)}, journal = {SIGPLAN Notices}, volume = {38}, pages = {263--274}, publisher = {ACM Press}, year = {2003}, month = aug, doi = {10.1145/944705.944729}, url = {https://doi.org/10.1145/944705.944729} }
2002
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Publication type: Conference paper
Automating Fraenkel-Mostowski Syntax
Track B Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002) eds. Víctor A. Carreño, César A. Muñoz, Sofiène Tahar NASA Conference Publication vol. NASA/CP-2002-211736 pp. 60–70 Hampton, Virginia, USA: National Aeronautics and Space Administration, Langley Research Center
- Author PDF for Automating Fraenkel-Mostowski Syntax
- NASA proceedings PDF for Automating Fraenkel-Mostowski Syntax
- BibTeX for Automating Fraenkel-Mostowski Syntax @inproceedings{gabbay:autfms, author = {Murdoch J. Gabbay}, title = {{Automating Fraenkel-Mostowski Syntax}}, booktitle = {Track B Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002)}, editor = {V{\'i}ctor A. Carre{\~n}o and C{\'e}sar A. Mu{\~n}oz and Sofi{\`e}ne Tahar}, series = {NASA Conference Publication}, volume = {NASA/CP-2002-211736}, pages = {60--70}, publisher = {National Aeronautics and Space Administration, Langley Research Center}, address = {Hampton, Virginia, USA}, year = {2002}, month = aug, url = {https://ntrs.nasa.gov/api/citations/20020063594/downloads/20020063594.pdf} }
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Publication type: Journal article
A New Approach to Abstract Syntax with Variable Binding
Formal Aspects of Computing vol. 13, no. 3–5 pp. 341–363
The paper that started it all, at least for me. We introduce nominal sets, atoms-abstraction, the NEW quantifier, and nominal abstract syntax (inductive datatypes of syntax-with-binding) and their inductive reasoning principles. See also the conference version in LICS99.
- Author PDF for A New Approach to Abstract Syntax with Variable Binding
- Publisher / DOI for A New Approach to Abstract Syntax with Variable Binding
- BibTeX for A New Approach to Abstract Syntax with Variable Binding @article{gabbay:newaas-jv, author = {Murdoch J. Gabbay and Andrew M. Pitts}, title = {{A New Approach to Abstract Syntax with Variable Binding}}, journal = {Formal Aspects of Computing}, volume = {13}, number = {3--5}, pages = {341--363}, year = {2002}, month = jul, doi = {10.1007/s001650200016}, url = {https://doi.org/10.1007/s001650200016} }
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Publication type: Conference paper
FM-HOL, a higher-order theory of names
Informal Proceedings of the Workshop on Thirty Five Years of Automath ed. Fairouz Kamareddine Edinburgh, Scotland: Heriot-Watt University
A generalisation of Fraenkel-Mostowski techniques to deal with infinite support and infinite atoms-abstraction. Largely but not completely superseded by a general mathematics of names.
- Author PDF for FM-HOL, a higher-order theory of names
- Workshop PDF for FM-HOL, a higher-order theory of names
- BibTeX for FM-HOL, a higher-order theory of names @inproceedings{gabbay:fmhotn, author = {Murdoch J. Gabbay}, title = {{FM-HOL, a higher-order theory of names}}, booktitle = {Informal Proceedings of the Workshop on Thirty Five Years of Automath}, editor = {Fairouz Kamareddine}, publisher = {Heriot-Watt University}, address = {Edinburgh, Scotland}, year = {2002}, month = apr, url = {https://www.macs.hw.ac.uk/~fairouz/forest/events/automath2002/informal-proceedings/ACC/gabbay.pdf} }
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Publication type: Workshop abstract
Nominal Sets, Equivariance Reasoning, and Variable Binding
Second Workshop on Meta-Logic (MLG 2002) 2 pp. total Kinosaki, Japan
A prose account of my talk at MLG02 in Kinosaki, Japan.
- Author PDF for Nominal Sets, Equivariance Reasoning, and Variable Binding
- BibTeX for Nominal Sets, Equivariance Reasoning, and Variable Binding @inproceedings{gabbay:nomser, author = {Murdoch J. Gabbay}, title = {{Nominal Sets, Equivariance Reasoning, and Variable Binding}}, booktitle = {Second Workshop on Meta-Logic (MLG 2002)}, address = {Kinosaki, Japan}, year = {2002}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/nomser.pdf} }
- Catalogue record for Nominal Sets, Equivariance Reasoning, and Variable Binding
2001
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Publication type: Doctoral thesis
A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
University of Cambridge Cambridge, UK
See also the survey paper on foundations on nominal techniques.
- Author PDF for A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
- BibTeX for A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language @phdthesis{gabbay:thesis, author = {Murdoch J. Gabbay}, title = {{A Theory of Inductive Definitions with {$\alpha$}-Equivalence: Semantics, Implementation, Programming Language}}, school = {University of Cambridge}, address = {Cambridge, UK}, year = {2001}, month = mar, url = {https://www.gabbay.org.uk/papers/thesis.pdf} }
2000
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Publication type: Conference paper
A Metalanguage for Programming with Bound Names Modulo Renaming
Proceedings of the 5th international conference on the mathematics of program construction (MPC 2000) Lecture Notes in Computer Science vol. 1837 pp. 230–255 Springer
- Author PDF for A Metalanguage for Programming with Bound Names Modulo Renaming
- Publisher / DOI for A Metalanguage for Programming with Bound Names Modulo Renaming
- BibTeX for A Metalanguage for Programming with Bound Names Modulo Renaming @inproceedings{gabbay:metpbn, author = {Andrew M. Pitts and Murdoch J. Gabbay}, title = {{A Metalanguage for Programming with Bound Names Modulo Renaming}}, booktitle = {Proceedings of the 5th international conference on the mathematics of program construction (MPC 2000)}, series = {Lecture Notes in Computer Science}, volume = {1837}, pages = {230--255}, publisher = {Springer}, year = {2000}, month = dec, doi = {10.1007/10722010_15}, url = {https://doi.org/10.1007/10722010_15} }
1999
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Publication type: Conference paper
A New Approach to Abstract Syntax Involving Binders
Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS 1999) pp. 214–224 IEEE Computer Society Press
My first conference paper. See also the journal version.
- Author PDF for A New Approach to Abstract Syntax Involving Binders
- Publisher / DOI for A New Approach to Abstract Syntax Involving Binders
- BibTeX for A New Approach to Abstract Syntax Involving Binders @inproceedings{gabbay:newaas, author = {Murdoch J. Gabbay and Andrew M. Pitts}, title = {{A New Approach to Abstract Syntax Involving Binders}}, booktitle = {Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS 1999)}, pages = {214--224}, publisher = {IEEE Computer Society Press}, year = {1999}, month = jul, doi = {10.1109/LICS.1999.782617}, url = {https://doi.org/10.1109/LICS.1999.782617} }
Publications by topic
Some publications appear under more than one topic.
Cryptography 1 publication
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Publication type: Journal article
Arithmetising Logic: Polynomial Semantics of FOL
Journal of Applied Logics vol. 12, no. 6 pp. 1481–1548 London: College Publications ISSN 2631-9810
- Author PDF for Arithmetising Logic: Polynomial Semantics of FOL
- Journal issue PDF for Arithmetising Logic: Polynomial Semantics of FOL
- BibTeX for Arithmetising Logic: Polynomial Semantics of FOL @article{gabbay:arithmetising-logic, author = {Murdoch J. Gabbay}, title = {{Arithmetising Logic: Polynomial Semantics of FOL}}, journal = {Journal of Applied Logics}, volume = {12}, number = {6}, pages = {1481--1548}, publisher = {College Publications}, address = {London}, year = {2025}, month = oct, issn = {2631-9810}, eprint = {2024/954}, url = {https://www.collegepublications.co.uk/downloads/ifcolog00074.pdf} }
- Repository record for Arithmetising Logic: Polynomial Semantics of FOL
Decentralised systems 12 publications
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Publication type: Technical report
Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
Anoma Research Topics Zenodo
- Author PDF for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- Repository for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- DOI 10.5281/zenodo.21074359 for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- BibTeX for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers @techreport{gabbay-sheff:consistent-histories, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2026}, month = jul, doi = {10.5281/zenodo.21074359}, url = {https://doi.org/10.5281/zenodo.21074359} }
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Publication type: Book chapter
Decentralized Collaborative Action: Cryptoeconomics in Space
Cryptoeconomic Theory eds. Melanie Swan, Soichiro Takagi, Frank Witte Advances in Science and Technology in the Age of AI vol. 1 pp. 53–70 Singapore: World Scientific ISBN 978-1-80061-865-7 eISBN 978-1-80061-867-1
- Author PDF for Decentralized Collaborative Action: Cryptoeconomics in Space
- Publisher / DOI for Decentralized Collaborative Action: Cryptoeconomics in Space
- BibTeX for Decentralized Collaborative Action: Cryptoeconomics in Space @incollection{gabbay:cryptoeconomics-in-space, author = {Murdoch J. Gabbay}, title = {{Decentralized Collaborative Action: Cryptoeconomics in Space}}, booktitle = {Cryptoeconomic Theory}, editor = {Melanie Swan and Soichiro Takagi and Frank Witte}, series = {Advances in Science and Technology in the Age of AI}, volume = {1}, pages = {53--70}, publisher = {World Scientific}, address = {Singapore}, year = {2026}, month = may, isbn = {978-1-80061-865-7}, note = {Electronic ISBN 978-1-80061-867-1}, doi = {10.1142/9781800618664_0004}, eprint = {2504.12493}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.1142/9781800618664_0004} }
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Publication type: Journal article
Semiframes: the algebra of semitopologies and actionable coalitions
Mathematical Structures in Computer Science vol. 36 article e3 Cambridge University Press
- Author PDF for Semiframes: the algebra of semitopologies and actionable coalitions
- Publisher / DOI for Semiframes: the algebra of semitopologies and actionable coalitions
- BibTeX for Semiframes: the algebra of semitopologies and actionable coalitions @article{gabbay:semiframes, author = {Murdoch J. Gabbay}, title = {{Semiframes: the algebra of semitopologies and actionable coalitions}}, journal = {Mathematical Structures in Computer Science}, volume = {36}, eid = {e3}, publisher = {Cambridge University Press}, year = {2026}, month = feb, note = {Article e3}, doi = {10.1017/S096012952510042X}, eprint = {2310.00956}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1017/S096012952510042X} }
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Technical report
Heterogeneous trust in reliable broadcast via modal logic and history structures
Anoma Research Topics Zenodo
- Author PDF for Heterogeneous trust in reliable broadcast via modal logic and history structures
- Repository for Heterogeneous trust in reliable broadcast via modal logic and history structures
- DOI 10.5281/zenodo.17636313 for Heterogeneous trust in reliable broadcast via modal logic and history structures
- BibTeX for Heterogeneous trust in reliable broadcast via modal logic and history structures @techreport{gabbay-sheff:heterogeneous-bracha-broadcast, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Heterogeneous trust in reliable broadcast via modal logic and history structures}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2025}, month = nov, doi = {10.5281/zenodo.17636313}, url = {https://doi.org/10.5281/zenodo.17636313} }
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Publication type: Journal article
Semitopology: a topological approach to decentralized collaborative action
Journal of Logic and Computation vol. 35, no. 5 article exae050 Oxford University Press online
- Author PDF for Semitopology: a topological approach to decentralized collaborative action
- Publisher / DOI for Semitopology: a topological approach to decentralized collaborative action
- BibTeX for Semitopology: a topological approach to decentralized collaborative action @article{gabbay:semitopology-jlc, author = {Murdoch J. Gabbay}, title = {{Semitopology: a topological approach to decentralized collaborative action}}, journal = {Journal of Logic and Computation}, volume = {35}, number = {5}, eid = {exae050}, publisher = {Oxford University Press}, year = {2025}, month = jul, note = {Article exae050}, doi = {10.1093/logcom/exae050}, eprint = {2303.09287}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1093/logcom/exae050} }
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Publication type: Preprint
A declarative approach to specifying Paxos using three-valued modal fixpoint logic
arXiv latest revision
- Author PDF for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- arXiv for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- BibTeX for A declarative approach to specifying Paxos using three-valued modal fixpoint logic @misc{gabbay-zanolini:declarative-distributed-three-valued, author = {Murdoch J. Gabbay and Luca Zanolini}, title = {{A declarative approach to specifying Paxos using three-valued modal fixpoint logic}}, howpublished = {arXiv preprint arXiv:2502.00892}, year = {2025}, month = feb, doi = {10.48550/arXiv.2502.00892}, eprint = {2502.00892}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2502.00892} }
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Publication type: Book
Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
Studies in Logic vol. 108 320 pp. total London: College Publications edition 1 ISBN 978-1-84890-465-1
- Author PDF for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- Publisher page for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- BibTeX for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic @book{gabbay:semitopology-monograph, author = {Murdoch J. Gabbay}, title = {{Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic}}, volume = {108}, publisher = {College Publications}, address = {London}, edition = {1}, year = {2024}, month = aug, isbn = {978-1-84890-465-1}, note = {320 pages}, eprint = {2402.03253}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://www.collegepublications.co.uk/logic/?00056=} }
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Publication type: Journal article
Algebras of UTxO blockchains
Mathematical Structures in Computer Science vol. 31, no. 9 In Homage to Martin Hofmann pp. 1034–1089 Cambridge University Press issue dated online
We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.
- Author PDF for Algebras of UTxO blockchains
- Publisher / DOI for Algebras of UTxO blockchains
- BibTeX for Algebras of UTxO blockchains @article{gabbay:algub, author = {Murdoch J. Gabbay}, title = {{Algebras of UTxO blockchains}}, journal = {Mathematical Structures in Computer Science}, volume = {31}, number = {9}, pages = {1034--1089}, publisher = {Cambridge University Press}, year = {2022}, month = jan, note = {Special issue: In Homage to Martin Hofmann}, doi = {10.1017/S0960129521000438}, url = {https://doi.org/10.1017/S0960129521000438} }
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Publication type: Conference paper
Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
3rd International Workshop on Formal Methods for Blockchains (FMBC 2021) eds. Bruno Bernardo, Diego Marmsoler Open Access Series in Informatics (OASIcs) vol. 95 pp. 2:1–2:14 article 2 Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik ISBN 978-3-95977-209-9 ISSN 2190-6807
- Author PDF for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
- Publisher / DOI for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard
- BibTeX for Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard @inproceedings{gabbay-jakobsson-sojakova:money-grows-proof-trees, author = {Murdoch J. Gabbay and Arvid Jakobsson and Kristina Sojakova}, title = {{Money Grows on (Proof-)Trees: The Formal FA1.2 Ledger Standard}}, booktitle = {3rd International Workshop on Formal Methods for Blockchains (FMBC 2021)}, editor = {Bruno Bernardo and Diego Marmsoler}, series = {Open Access Series in Informatics (OASIcs)}, volume = {95}, pages = {2:1--2:14}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, year = {2021}, month = nov, isbn = {978-3-95977-209-9}, issn = {2190-6807}, doi = {10.4230/OASIcs.FMBC.2021.2}, url = {https://doi.org/10.4230/OASIcs.FMBC.2021.2} }
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Publication type: Book chapter
Karl Marx and the Blockchain
Media, Technology and Education in a Post-Truth Society ed. Grech, Alex pp. 225–241 Emerald Publishing Limited, Bingley
We consider blockchain technology from a historical, social, and economic perspective, through a lens of the economic theories of Karl Marx.
- Author PDF for Karl Marx and the Blockchain
- Publisher / DOI for Karl Marx and the Blockchain
- BibTeX for Karl Marx and the Blockchain @incollection{gabbay:karmb, author = {Devraj Basu and Murdoch J. Gabbay}, title = {{Karl Marx and the Blockchain}}, booktitle = {Media, Technology and Education in a Post-Truth Society}, pages = {225--241}, publisher = {Emerald Publishing Limited, Bingley}, year = {2021}, month = jul, doi = {10.1108/978-1-80043-906-120211016}, url = {https://doi.org/10.1108/978-1-80043-906-120211016} }
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Publication type: Conference paper
UTxO- vs account-based smart contract blockchain programming paradigms
Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020) eds. Tiziana Margaria, Bernhard Steffen Lecture Notes in Computer Science vol. 12478 pp. 73–88 Springer
We implement two versions of a simple but illustrative smart contract: one in Solidity on the Ethereum blockchain platform, and one in Plutus on the Cardano platform, with annotated code excerpts and with source code attached. We prove some simple but novel results about observational equivalence and conclude with a wide-ranging and detailed discussion in the light of the examples, mathematical model, and mathematical results so far.
- Author PDF for UTxO- vs account-based smart contract blockchain programming paradigms
- Publisher / DOI for UTxO- vs account-based smart contract blockchain programming paradigms
- BibTeX for UTxO- vs account-based smart contract blockchain programming paradigms @inproceedings{gabbay:utxabs, author = {Lars Br{\"u}njes and Murdoch J. Gabbay}, title = {{UTxO- vs account-based smart contract blockchain programming paradigms}}, booktitle = {Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020)}, editor = {Tiziana Margaria and Bernhard Steffen}, series = {Lecture Notes in Computer Science}, volume = {12478}, pages = {73--88}, publisher = {Springer}, year = {2020}, month = oct, doi = {10.1007/978-3-030-61467-6_6}, eprint = {2003.14271}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1007/978-3-030-61467-6_6} }
- arXiv preprint for UTxO- vs account-based smart contract blockchain programming paradigms
Declarative distributed protocols 4 publications
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Publication type: Technical report
Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
Anoma Research Topics Zenodo
- Author PDF for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- Repository for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- DOI 10.5281/zenodo.21074359 for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers
- BibTeX for Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers @techreport{gabbay-sheff:consistent-histories, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Consistent histories: enforcing linearity and excluding double-spending amongst collaborating controllers}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2026}, month = jul, doi = {10.5281/zenodo.21074359}, url = {https://doi.org/10.5281/zenodo.21074359} }
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Technical report
Heterogeneous trust in reliable broadcast via modal logic and history structures
Anoma Research Topics Zenodo
- Author PDF for Heterogeneous trust in reliable broadcast via modal logic and history structures
- Repository for Heterogeneous trust in reliable broadcast via modal logic and history structures
- DOI 10.5281/zenodo.17636313 for Heterogeneous trust in reliable broadcast via modal logic and history structures
- BibTeX for Heterogeneous trust in reliable broadcast via modal logic and history structures @techreport{gabbay-sheff:heterogeneous-bracha-broadcast, author = {Murdoch J. Gabbay and Isaac Sheff}, title = {{Heterogeneous trust in reliable broadcast via modal logic and history structures}}, institution = {Anoma Research Topics}, publisher = {Zenodo}, year = {2025}, month = nov, doi = {10.5281/zenodo.17636313}, url = {https://doi.org/10.5281/zenodo.17636313} }
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Publication type: Preprint
A declarative approach to specifying Paxos using three-valued modal fixpoint logic
arXiv latest revision
- Author PDF for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- arXiv for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- BibTeX for A declarative approach to specifying Paxos using three-valued modal fixpoint logic @misc{gabbay-zanolini:declarative-distributed-three-valued, author = {Murdoch J. Gabbay and Luca Zanolini}, title = {{A declarative approach to specifying Paxos using three-valued modal fixpoint logic}}, howpublished = {arXiv preprint arXiv:2502.00892}, year = {2025}, month = feb, doi = {10.48550/arXiv.2502.00892}, eprint = {2502.00892}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2502.00892} }
Lambda calculus and metaprogramming 15 publications
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Publication type: Journal article
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
Annals of Pure and Applied Logic vol. 168, no. 3 pp. 501–621 Elsevier
Everything you wanted to know about semantics of the lambda-calculus using nominal techniques.
- Author PDF for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- Publisher / DOI for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- BibTeX for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness @article{gabbay:repdul, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness}}, journal = {Annals of Pure and Applied Logic}, volume = {168}, number = {3}, pages = {501--621}, publisher = {Elsevier}, year = {2017}, month = mar, doi = {10.1016/j.apal.2016.10.001}, url = {https://doi.org/10.1016/j.apal.2016.10.001} }
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Publication type: Journal article
A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
The IfCoLog Journal of Logics and their Applications vol. 1, no. 2 pp. 83–106 College Publications ISBN 978-1-84890-163-6 ISSN 2055-3706
- Repository PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- Journal issue PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- DOI 10.17863/CAM.42150 for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- BibTeX for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality @article{gabbay:simcmt, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality}}, journal = {The IfCoLog Journal of Logics and their Applications}, volume = {1}, number = {2}, pages = {83--106}, publisher = {College Publications}, year = {2014}, month = nov, isbn = {978-1-84890-163-6}, issn = {2055-3706}, doi = {10.17863/CAM.42150}, url = {https://doi.org/10.17863/CAM.42150} }
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Publication type: Journal article
Denotation of contextual modal type theory (CMTT): syntax and meta-programming
Journal of Applied Logic vol. 11, no. 1 pp. 1–29
A denotation for CMTT. I wanted to understand this for years, and now we have an answer!
- Author PDF for Denotation of contextual modal type theory (CMTT): syntax and meta-programming
- Publisher / DOI for Denotation of contextual modal type theory (CMTT): syntax and meta-programming
- BibTeX for Denotation of contextual modal type theory (CMTT): syntax and meta-programming @article{gabbay:dencmt, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of contextual modal type theory (CMTT): syntax and meta-programming}}, journal = {Journal of Applied Logic}, volume = {11}, number = {1}, pages = {1--29}, year = {2013}, month = mar, doi = {10.1016/j.jal.2012.07.002}, url = {https://doi.org/10.1016/j.jal.2012.07.002} }
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Publication type: Preprint
Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
Subsumed by “Denotation of contextual modal type theory (CMTT): syntax and meta-programming”.
- Author PDF for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- arXiv record (1202.0904) for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- BibTeX for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT) @misc{gabbay:densmc, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)}}, howpublished = {arXiv preprint arXiv:1202.0904}, year = {2012}, month = feb, eprint = {1202.0904}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/1202.0904} }
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Publication type: Conference paper
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011) EPTCS vol. 71 pp. 58–75
Henkin-style models of the lambda-calculus in nominal sets. No Tarski-style valuations; instead we use nominal substitution algebras. The title of the paper in the pre-proceedings was Semantics for functions of the simply-typed lambda-calculus with nominal sets.
- Author PDF for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- Publisher / DOI for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- BibTeX for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets @inproceedings{gabbay:nomhss, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets}}, booktitle = {Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011)}, series = {EPTCS}, volume = {71}, pages = {58--75}, year = {2011}, month = sep, doi = {10.4204/EPTCS.71.5}, url = {https://doi.org/10.4204/EPTCS.71.5} }
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Publication type: Conference paper
A simple class of Kripke-style models in which logic and computation have equal standing
International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010) eds. Edmund M. Clarke, Andrei Voronkov Lecture Notes in Computer Science vol. 6355 pp. 231–254 Springer
A sound and complete model of lambda-calculus reductions in structures related to Kripke structures. We construct a logic for the models and identify lambda-terms with predicates by direct compositional translation. Reduction becomes logical entailment.
- Author PDF for A simple class of Kripke-style models in which logic and computation have equal standing
- Publisher / DOI for A simple class of Kripke-style models in which logic and computation have equal standing
- BibTeX for A simple class of Kripke-style models in which logic and computation have equal standing @inproceedings{gabbay:simcks, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A simple class of Kripke-style models in which logic and computation have equal standing}}, booktitle = {International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010)}, editor = {Edmund M. Clarke and Andrei Voronkov}, series = {Lecture Notes in Computer Science}, volume = {6355}, pages = {231--254}, publisher = {Springer}, year = {2010}, month = dec, doi = {10.1007/978-3-642-17511-4_14}, url = {https://doi.org/10.1007/978-3-642-17511-4_14} }
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Publication type: Journal article
A nominal axiomatisation of the lambda-calculus
Journal of Logic and Computation vol. 20, no. 2 pp. 501–531
We can write axioms for the lambda-calculus in nominal algebra—but do we know those axioms are right? We prove completeness of axioms for alpha-beta-(eta) equivalence. This subsumes a previous book chapter.
- Author PDF for A nominal axiomatisation of the lambda-calculus
- Publisher / DOI for A nominal axiomatisation of the lambda-calculus
- BibTeX for A nominal axiomatisation of the lambda-calculus @article{gabbay:nomalc, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{A nominal axiomatisation of the lambda-calculus}}, journal = {Journal of Logic and Computation}, volume = {20}, number = {2}, pages = {501--531}, year = {2010}, month = apr, doi = {10.1093/logcom/exp049}, url = {https://doi.org/10.1093/logcom/exp049} }
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Publication type: Journal article
Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
Information and Computation vol. 208, no. 3 pp. 230–258
Types-as-propositions, incomplete (nominal) terms as incomplete derivations. That is, the “variables” of nominal terms become “holes” in first-order logic derivations. A freshness condition becomes a promise that an assumption will be discharged.
This journal paper expands on the conference version.
- Author PDF for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- Publisher / DOI for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- BibTeX for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms @article{gabbay:curhif-jv, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms}}, journal = {Information and Computation}, volume = {208}, number = {3}, pages = {230--258}, year = {2010}, month = mar, doi = {10.1016/j.ic.2009.09.003}, url = {https://doi.org/10.1016/j.ic.2009.09.003} }
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Publication type: Journal article
The lambda-context calculus (extended version)
Information and computation vol. 207, no. 12 pp. 1369–1400 Elsevier
The lambda-context calculus is a multi-level lambda-calculus, generalising a two-level lambda-calculus with an infinite hierarchy of variables such that higher level variables behave like meta-variables with respect to lower level variables. It can express both capture-avoiding and capturing substitution (instantiation), explicit substitutions, references, and more.
An extended version of our LFMTP’07 paper.
- Author PDF for The lambda-context calculus (extended version)
- Publisher / DOI for The lambda-context calculus (extended version)
- BibTeX for The lambda-context calculus (extended version) @article{gabbay:lamcce, author = {Murdoch J. Gabbay and St{\'e}phane Lengrand}, title = {{The lambda-context calculus (extended version)}}, journal = {Information and computation}, volume = {207}, number = {12}, pages = {1369--1400}, publisher = {Elsevier}, year = {2009}, month = dec, doi = {10.1016/j.ic.2009.06.004}, url = {https://doi.org/10.1016/j.ic.2009.06.004} }
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Publication type: Conference paper
Two-level lambda-calculus
Electronic Notes in Theoretical Computer Science vol. 246 pp. 107–129
The two-level lambda-calculus gives a functional operational semantics to nominal terms unknowns; the result is a lambda-calculus with capturing and capture-avoiding substitution are represented and nominal terms style alpha-equivalence for level 1 variables (atoms) in the presence of level 2 variables (unknowns).
This ENTCS paper subsumes a 15 page workshop paper entitled “Two-and-a-halfth order lambda-calculus”, which appeared in the preproceedings of WFLP08.
- Author PDF for Two-level lambda-calculus
- Publisher / DOI for Two-level lambda-calculus
- BibTeX for Two-level lambda-calculus @inproceedings{gabbay:twollc, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Two-level lambda-calculus}}, booktitle = {Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008)}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {246}, pages = {107--129}, year = {2009}, month = aug, doi = {10.1016/j.entcs.2009.07.018}, url = {https://doi.org/10.1016/j.entcs.2009.07.018} }
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Publication type: Conference paper
Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
Proceedings of the 4th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2009) pp. 64–73 ACM
We translate between equality derivations in lambda-theories and equality derivations in permissive-nominal algebra theories. This connects reasoning over higher-order syntax to reasoning over nominal terms.
- Author PDF for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
- Publisher / DOI for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables
- BibTeX for Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables @inproceedings{gabbay:unialt, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Universal algebra over lambda-terms and nominal terms: the connection in logic between nominal techniques and higher-order variables}}, booktitle = {Proceedings of the 4th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2009)}, pages = {64--73}, publisher = {ACM}, year = {2009}, month = aug, doi = {10.1145/1577824.1577835}, url = {https://doi.org/10.1145/1577824.1577835} }
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Publication type: Book chapter
The lambda-calculus is nominal algebraic
Reasoning in Simple Type Theory: Festschrift in Honor of Peter B. Andrews on his 70th Birthday eds. Christoph Benzmüller, Chad E. Brown, Jörg Siekmann, Richard Statman Studies in Logic vol. 17 pp. 271–302 London: College Publications ISBN 978-1-904987-70-3
We use nominal algebra to give a sound and complete axiomatisation of the lambda-calculus. Subsumed by a journal version.
- Author PDF for The lambda-calculus is nominal algebraic
- Publisher page for The lambda-calculus is nominal algebraic
- BibTeX for The lambda-calculus is nominal algebraic @incollection{gabbay:lamcna, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{The lambda-calculus is nominal algebraic}}, booktitle = {Reasoning in Simple Type Theory: Festschrift in Honor of Peter B. Andrews on his 70th Birthday}, editor = {Christoph Benzm{\"u}ller and Chad E. Brown and J{\"o}rg Siekmann and Richard Statman}, series = {Studies in Logic}, volume = {17}, pages = {271--302}, publisher = {College Publications}, address = {London}, year = {2008}, month = dec, isbn = {978-1-904987-70-3}, url = {https://www.collegepublications.co.uk/logic/mlf/?00010=} }
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Publication type: Conference paper
One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008) Lecture Notes in Artificial Intelligence vol. 5110 pp. 179–193 Springer
Nominal terms give Curry-Howard for schematic derivations. Atoms correspond with assumptions, unknowns correspond with schematic parts of the derivation, freshness conditions corresponds with the condition that an assumption will be discharged (even if the schema does not specify how).
- Author PDF for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- Publisher / DOI for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- BibTeX for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations @inproceedings{gabbay:curhid, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations}}, booktitle = {Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008)}, series = {Lecture Notes in Artificial Intelligence}, volume = {5110}, pages = {179--193}, publisher = {Springer}, year = {2008}, month = jun, doi = {10.1007/978-3-540-69937-8_16}, url = {https://doi.org/10.1007/978-3-540-69937-8_16} }
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Publication type: Journal article
The lambda-context calculus
Electronic Notes in Theoretical Computer Science vol. 196 pp. 19–35 Elsevier ISSN 1571-0661
We present a simple lambda-calculus whose syntax is populated by variables which behave like meta-variables. It can express both capture-avoiding and capturing substitution (instantiation).
- Author PDF for The lambda-context calculus
- Publisher / DOI for The lambda-context calculus
- BibTeX for The lambda-context calculus @article{gabbay:lamcc, author = {Murdoch J. Gabbay and St{\'e}phane Lengrand}, title = {{The lambda-context calculus}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {196}, pages = {19--35}, publisher = {Elsevier}, year = {2008}, month = jan, doi = {10.1016/j.entcs.2007.09.015}, url = {https://doi.org/10.1016/j.entcs.2007.09.015} }
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Publication type: Conference paper
A NEW calculus of contexts
Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05) eds. Pedro Barahona, Amy P. Felty pp. 94–105 Lisbon, Portugal: Association for Computing Machinery ISBN 1-59593-090-6
Subsumed by a later journal paper. A lambda-calculus with context substitution (which does not avoid capture) alongside ordinary beta-reduction. It can encode dynamic binding, objects, and interesting properties associated to partial evaluation. Meta-properties include an applicative characterisation of contextual equivalence.
- Author PDF for A NEW calculus of contexts
- Publisher / DOI for A NEW calculus of contexts
- BibTeX for A NEW calculus of contexts @inproceedings{gabbay:newcc, author = {Murdoch J. Gabbay}, title = {{A NEW calculus of contexts}}, booktitle = {Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05)}, editor = {Pedro Barahona and Amy P. Felty}, pages = {94--105}, publisher = {Association for Computing Machinery}, address = {Lisbon, Portugal}, year = {2005}, month = jul, isbn = {1-59593-090-6}, doi = {10.1145/1069774.1069783}, url = {https://doi.org/10.1145/1069774.1069783} }
Logic and proof theory 33 publications
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Publication type: Preprint
Cryptographic certificates of validity for trustworthy AI
- Author PDF for Cryptographic certificates of validity for trustworthy AI
- arXiv for Cryptographic certificates of validity for trustworthy AI
- BibTeX for Cryptographic certificates of validity for trustworthy AI @misc{gabbay:cryptographic-certificates-ai, author = {Murdoch J. Gabbay}, title = {{Cryptographic certificates of validity for trustworthy AI}}, howpublished = {arXiv preprint arXiv:2606.23768}, year = {2026}, month = jun, doi = {10.48550/arXiv.2606.23768}, eprint = {2606.23768}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.48550/arXiv.2606.23768} }
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Journal article
Arithmetising Logic: Polynomial Semantics of FOL
Journal of Applied Logics vol. 12, no. 6 pp. 1481–1548 London: College Publications ISSN 2631-9810
- Author PDF for Arithmetising Logic: Polynomial Semantics of FOL
- Journal issue PDF for Arithmetising Logic: Polynomial Semantics of FOL
- BibTeX for Arithmetising Logic: Polynomial Semantics of FOL @article{gabbay:arithmetising-logic, author = {Murdoch J. Gabbay}, title = {{Arithmetising Logic: Polynomial Semantics of FOL}}, journal = {Journal of Applied Logics}, volume = {12}, number = {6}, pages = {1481--1548}, publisher = {College Publications}, address = {London}, year = {2025}, month = oct, issn = {2631-9810}, eprint = {2024/954}, url = {https://www.collegepublications.co.uk/downloads/ifcolog00074.pdf} }
- Repository record for Arithmetising Logic: Polynomial Semantics of FOL
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Publication type: Preprint
A declarative approach to specifying Paxos using three-valued modal fixpoint logic
arXiv latest revision
- Author PDF for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- arXiv for A declarative approach to specifying Paxos using three-valued modal fixpoint logic
- BibTeX for A declarative approach to specifying Paxos using three-valued modal fixpoint logic @misc{gabbay-zanolini:declarative-distributed-three-valued, author = {Murdoch J. Gabbay and Luca Zanolini}, title = {{A declarative approach to specifying Paxos using three-valued modal fixpoint logic}}, howpublished = {arXiv preprint arXiv:2502.00892}, year = {2025}, month = feb, doi = {10.48550/arXiv.2502.00892}, eprint = {2502.00892}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2502.00892} }
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Publication type: Journal article
Quantifiers in logic and proof-search using permissive-nominal terms and sets
Journal of Logic and Computation vol. 25, no. 2 pp. 473–523
Nominal terms used to model the γ- and δ-rules controlling quantifiers in reductive proof search.
- Author PDF for Quantifiers in logic and proof-search using permissive-nominal terms and sets
- Publisher / DOI for Quantifiers in logic and proof-search using permissive-nominal terms and sets
- BibTeX for Quantifiers in logic and proof-search using permissive-nominal terms and sets @article{gabbay:qualps, author = {Murdoch J. Gabbay and Claus-Peter Wirth}, title = {{Quantifiers in logic and proof-search using permissive-nominal terms and sets}}, journal = {Journal of Logic and Computation}, volume = {25}, number = {2}, pages = {473--523}, year = {2015}, month = apr, doi = {10.1093/logcom/exs057}, url = {https://doi.org/10.1093/logcom/exs057} }
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Publication type: Book chapter
Nominal terms and nominal logics: from foundations to meta-mathematics
Handbook of Philosophical Logic vol. 17 pp. 79–178 Springer
A survey and update chapter describing a journey from (permissive-)nominal sets, via nominal terms, rewriting and algebra, to permissive-nominal logic and a finite first-order nominal axiomatisation of arithmetic. See also Foundations of nominal techniques.
- Author PDF for Nominal terms and nominal logics: from foundations to meta-mathematics
- Publisher / DOI for Nominal terms and nominal logics: from foundations to meta-mathematics
- BibTeX for Nominal terms and nominal logics: from foundations to meta-mathematics @incollection{gabbay:nomtnl, author = {Murdoch J. Gabbay}, title = {{Nominal terms and nominal logics: from foundations to meta-mathematics}}, booktitle = {Handbook of Philosophical Logic}, volume = {17}, pages = {79--178}, publisher = {Springer}, year = {2013}, month = jan, doi = {10.1007/978-94-007-6600-6_2}, url = {https://doi.org/10.1007/978-94-007-6600-6_2} }
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Publication type: Journal article
PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
Theoretical Computer Science vol. 451 pp. 38–69
We translate PNL to HOL (permissive-nominal logic to higher-order logic). Notably, this involved turning a nominal atoms-abstraction into a total function.
- Author PDF for PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
- Publisher / DOI for PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
- BibTeX for PNL to HOL: from the logic of nominal sets to the logic of higher-order functions @article{gabbay:pnlthf, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{PNL to HOL: from the logic of nominal sets to the logic of higher-order functions}}, journal = {Theoretical Computer Science}, volume = {451}, pages = {38--69}, year = {2012}, month = sep, doi = {10.1016/j.tcs.2012.06.007}, url = {https://doi.org/10.1016/j.tcs.2012.06.007} }
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Publication type: Journal article
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
Journal of Symbolic Logic vol. 77, no. 3 pp. 828–852 Duke University Press
Moving between permissive-nominal (infinitely-supported) and nominal (finitely-supported) models, with applications to permissive-nominal algebra completeness and permissive-nominal logic completeness. Cool stuff playing with nominal sets.
- Author PDF for Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
- Publisher / DOI for Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
- BibTeX for Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free @article{gabbay:finisn, author = {Murdoch J. Gabbay}, title = {{Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free}}, journal = {Journal of Symbolic Logic}, volume = {77}, number = {3}, pages = {828--852}, publisher = {Duke University Press}, year = {2012}, month = sep, doi = {10.2178/jsl/1344862164}, url = {https://doi.org/10.2178/jsl/1344862164} }
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Publication type: Journal article
Permissive-nominal logic: First-order logic over nominal terms and sets
ACM Transactions on Computational Logic vol. 13, no. 3 pp. 1–36 article 20 New York: Association for Computing Machinery
First-order logic with term-formers that can bind names. So we can axiomatise things like forall or lambda, or finitely axiomatise arithmetic. This gives PNL much of the expressive power of higher-order logic: but terms, derivations and models of PNL are first-order and the logic seems to strike a good balance between expressivity and simplicity. In brief, PNL is obtained by adding universal quantification to nominal algebra. But, to make that work we needed to invent permission sets.
This is a journal version of the conference paper.
- Author PDF for Permissive-nominal logic: First-order logic over nominal terms and sets
- Publisher / DOI for Permissive-nominal logic: First-order logic over nominal terms and sets
- BibTeX for Permissive-nominal logic: First-order logic over nominal terms and sets @article{gabbay:pernl-jv, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Permissive-nominal logic: First-order logic over nominal terms and sets}}, journal = {ACM Transactions on Computational Logic}, volume = {13}, number = {3}, eid = {20}, pages = {1--36}, publisher = {Association for Computing Machinery}, year = {2012}, month = aug, doi = {10.1145/2287718.2287720}, url = {https://doi.org/10.1145/2287718.2287720} }
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Publication type: Journal article
Unity in nominal equational reasoning: The algebra of equality on nominal sets
Journal of Applied Logic vol. 10, no. 2 pp. 199–217
An exhaustive analysis of the design choices involved in nominal equational / equality / algebra logic over nominal sets.
- Author PDF for Unity in nominal equational reasoning: The algebra of equality on nominal sets
- Publisher / DOI for Unity in nominal equational reasoning: The algebra of equality on nominal sets
- BibTeX for Unity in nominal equational reasoning: The algebra of equality on nominal sets @article{gabbay:uniner, author = {Murdoch J. Gabbay}, title = {{Unity in nominal equational reasoning: The algebra of equality on nominal sets}}, journal = {Journal of Applied Logic}, volume = {10}, number = {2}, pages = {199--217}, year = {2012}, month = jun, doi = {10.1016/j.jal.2012.03.001}, url = {https://doi.org/10.1016/j.jal.2012.03.001} }
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Publication type: Workshop abstract
Quantifier rules in reductive proof using nominal semantics
Automated Reasoning Workshop (ARW 2012) 2 pp. total
We consider liberalised delta-rules in proof-search, and give them a nominal algebraic semantics.
- Author PDF for Quantifier rules in reductive proof using nominal semantics
- BibTeX for Quantifier rules in reductive proof using nominal semantics @inproceedings{gabbay:quarrp-ea, author = {Murdoch J. Gabbay}, title = {{Quantifier rules in reductive proof using nominal semantics}}, booktitle = {Automated Reasoning Workshop (ARW 2012)}, year = {2012}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/quarrp-ea.pdf} }
- Catalogue record for Quantifier rules in reductive proof using nominal semantics
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Publication type: Preprint
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Subsumed by “PNL to HOL: from the logic of nominal sets to the logic of higher-order functions”.
- Author PDF for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
- arXiv record (1111.4611) for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
- BibTeX for From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions @misc{gabbay:nomsbf-arxiv, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions}}, howpublished = {arXiv preprint arXiv:1111.4611}, year = {2011}, month = nov, eprint = {1111.4611}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/1111.4611} }
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Publication type: Workshop abstract
Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
Automated Reasoning Workshop (ARW 2011) 2 pp. total
We sketch what theorem-proving might look like if it were based on nominal terms. This eventually evolved into Nominal terms and nominal logics.
- Author PDF for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
- BibTeX for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover @inproceedings{gabbay:metbnt-ea, author = {Murdoch J. Gabbay}, title = {{Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover}}, booktitle = {Automated Reasoning Workshop (ARW 2011)}, year = {2011}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/metbnt-ea.pdf} }
- Catalogue record for Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
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Publication type: Conference paper
A simple class of Kripke-style models in which logic and computation have equal standing
International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010) eds. Edmund M. Clarke, Andrei Voronkov Lecture Notes in Computer Science vol. 6355 pp. 231–254 Springer
A sound and complete model of lambda-calculus reductions in structures related to Kripke structures. We construct a logic for the models and identify lambda-terms with predicates by direct compositional translation. Reduction becomes logical entailment.
- Author PDF for A simple class of Kripke-style models in which logic and computation have equal standing
- Publisher / DOI for A simple class of Kripke-style models in which logic and computation have equal standing
- BibTeX for A simple class of Kripke-style models in which logic and computation have equal standing @inproceedings{gabbay:simcks, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A simple class of Kripke-style models in which logic and computation have equal standing}}, booktitle = {International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010)}, editor = {Edmund M. Clarke and Andrei Voronkov}, series = {Lecture Notes in Computer Science}, volume = {6355}, pages = {231--254}, publisher = {Springer}, year = {2010}, month = dec, doi = {10.1007/978-3-642-17511-4_14}, url = {https://doi.org/10.1007/978-3-642-17511-4_14} }
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Publication type: Conference paper
Permissive Nominal Logic
Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010) pp. 165–176 New York: ACM Press
First-order logic with binding term-formers. We can axiomatise forall or lambda, and finitely axiomatise arithmetic. PNL has much of the expressive power of higher-order logic but terms, derivations and models of PNL are first-order. PNL is nominal algebra” + quantification over nominal unknowns. See also the journal version.
- Author PDF for Permissive Nominal Logic
- Publisher / DOI for Permissive Nominal Logic
- BibTeX for Permissive Nominal Logic @inproceedings{gabbay:pernl, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Permissive Nominal Logic}}, booktitle = {Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010)}, pages = {165--176}, publisher = {ACM Press}, year = {2010}, month = jul, doi = {10.1145/1836089.1836111}, url = {https://doi.org/10.1145/1836089.1836111} }
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Publication type: Journal article
Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
Information and Computation vol. 208, no. 3 pp. 230–258
Types-as-propositions, incomplete (nominal) terms as incomplete derivations. That is, the “variables” of nominal terms become “holes” in first-order logic derivations. A freshness condition becomes a promise that an assumption will be discharged.
This journal paper expands on the conference version.
- Author PDF for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- Publisher / DOI for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms
- BibTeX for Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms @article{gabbay:curhif-jv, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms}}, journal = {Information and Computation}, volume = {208}, number = {3}, pages = {230--258}, year = {2010}, month = mar, doi = {10.1016/j.ic.2009.09.003}, url = {https://doi.org/10.1016/j.ic.2009.09.003} }
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Publication type: Journal article
Nominal universal algebra: equational logic with names and binding
Journal of Logic and Computation vol. 19, no. 6 pp. 1455–1508
Nominal Algebra, a logic for equational reasoning over nominal sets, which uses nominal terms. Think: “algebra with term-formers that can bind”. This paper subsumes conference versions in WOLLIC07 and NWPT06, and a 2006 technical report, and is followed by Nominal terms, nominal logics.
Many of the papers applying nominal algebra were in print before the journal paper. You might be interested in
- Author PDF for Nominal universal algebra: equational logic with names and binding
- Publisher / DOI for Nominal universal algebra: equational logic with names and binding
- BibTeX for Nominal universal algebra: equational logic with names and binding @article{gabbay:nomuae, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal universal algebra: equational logic with names and binding}}, journal = {Journal of Logic and Computation}, volume = {19}, number = {6}, pages = {1455--1508}, year = {2009}, month = dec, doi = {10.1093/logcom/exp033}, url = {https://doi.org/10.1093/logcom/exp033} }
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Publication type: Conference paper
Term sequent logic
Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008) Electronic Notes in Theoretical Computer Science vol. 246 pp. 87–106
Term sequents are sequents for terms instead of predicates. We build lambda-calculus reductions, similar to rewriting but with structure corresponding to proof-theory. This subsumes a 15 page workshop paper of the same title which appeared in the preproceedings of WFLP08.
- Author PDF for Term sequent logic
- Publisher / DOI for Term sequent logic
- BibTeX for Term sequent logic @inproceedings{gabbay:tersl, author = {Michael Gabbay and Murdoch J. Gabbay}, title = {{Term sequent logic}}, booktitle = {Proceedings of the 17th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2008)}, series = {Electronic Notes in Theoretical Computer Science}, volume = {246}, pages = {87--106}, year = {2009}, month = aug, doi = {10.1016/j.entcs.2009.07.017}, url = {https://doi.org/10.1016/j.entcs.2009.07.017} }
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Publication type: Journal article
Nominal Algebra and the HSP Theorem
Journal of Logic and Computation vol. 19, no. 2 pp. 341–367
It is not obvious that Nominal Algebra is a pure equational logic, because of freshness conditions. E.g. η-conversion (eta-conversion) looks like this: a#X⊢λa.(Xa)=X. This is a conditional equality. The HSPA theorem for Nominal Algebra asserts that any model of a Nominal Algebra theory is part of a quotient of a product of atoms-abstractions of free models.
This limits the complexity of models, and puts a precise measure on the notion of equality of nominal algebra: making things equal, putting things into tuples, and forming atoms-abstracton.
- Author PDF for Nominal Algebra and the HSP Theorem
- Publisher / DOI for Nominal Algebra and the HSP Theorem
- BibTeX for Nominal Algebra and the HSP Theorem @article{gabbay:nomahs, author = {Murdoch J. Gabbay}, title = {{Nominal Algebra and the HSP Theorem}}, journal = {Journal of Logic and Computation}, volume = {19}, number = {2}, pages = {341--367}, year = {2009}, month = apr, doi = {10.1093/logcom/exn055}, url = {https://doi.org/10.1093/logcom/exn055} }
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Publication type: Journal article
One-and-a-halfth-order Logic
Journal of Logic and Computation vol. 18, no. 4 pp. 521–562
A logic of schematic reasoning based on nominal algebra. For instance, we can represent the schematic assertion “if a does not occur in φ then φ implies ∀a.φ” and its derivation as “a#X⊢X⊃∀a.X” and corresponding schematic derivations. We give algebraic and sequent characterisations, and show correctness with respect to first-order logic. This subsumes the PPDP06 conference paper.
- Author PDF for One-and-a-halfth-order Logic
- Publisher / DOI for One-and-a-halfth-order Logic
- BibTeX for One-and-a-halfth-order Logic @article{gabbay:oneaah-jv, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{One-and-a-halfth-order Logic}}, journal = {Journal of Logic and Computation}, volume = {18}, number = {4}, pages = {521--562}, year = {2008}, month = aug, doi = {10.1093/logcom/exm064}, url = {https://doi.org/10.1093/logcom/exm064} }
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Publication type: Journal article
a-logic with arrows (journal version)
Electronic Notes in Theoretical Computer Science vol. 216 pp. 3–29
a-logic has a predicate ‘atom’ which is true of a term when it is a variable symbol We prove cut-elimination, give a notion of model, prove soundess and completeness, and axiomatise substitution and the lambda-calculus, using ‘atom’ to identify variables. This is a journal version of our WFLP07 paper, with an improved proof-system.
- Author PDF for a-logic with arrows (journal version)
- Publisher / DOI for a-logic with arrows (journal version)
- BibTeX for a-logic with arrows (journal version) @article{gabbay:alwa-jv, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic with arrows (journal version)}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {216}, pages = {3--29}, year = {2008}, month = jul, doi = {10.1016/j.entcs.2008.06.031}, url = {https://doi.org/10.1016/j.entcs.2008.06.031} }
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Publication type: Conference paper
One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008) Lecture Notes in Artificial Intelligence vol. 5110 pp. 179–193 Springer
Nominal terms give Curry-Howard for schematic derivations. Atoms correspond with assumptions, unknowns correspond with schematic parts of the derivation, freshness conditions corresponds with the condition that an assumption will be discharged (even if the schema does not specify how).
- Author PDF for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- Publisher / DOI for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations
- BibTeX for One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations @inproceedings{gabbay:curhid, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{One-and-a-Halfth Order Terms: Curry-Howard and Incomplete Derivations}}, booktitle = {Proceedings of 15th Workshop on Logic, Language and Information in Computation (WoLLIC 2008)}, series = {Lecture Notes in Artificial Intelligence}, volume = {5110}, pages = {179--193}, publisher = {Springer}, year = {2008}, month = jun, doi = {10.1007/978-3-540-69937-8_16}, url = {https://doi.org/10.1007/978-3-540-69937-8_16} }
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Publication type: Workshop abstract
Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics 2 pp. total Paris, France
- Author PDF for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
- BibTeX for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants @inproceedings{gabbay:towpta, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants}}, booktitle = {Paris Arch{\'e} workshop on abstract objects in semantics and the philosophy of mathematics}, address = {Paris, France}, year = {2008}, month = feb, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/towpta.pdf} }
- Catalogue record for Towards a Proof-Theoretic Approach to Plurality: Numbers Are Logical Constants
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Publication type: Technical report
Nominal Algebra and the HSP Theorem
Heriot-Watt University HW-MACS-TR-0057
Subsumed by “Nominal Algebra and the HSP Theorem”.
We prove the HSP (Homomorphism, Subalgebra, Product) theorem for Nominal Algebra, also known as Birkhoffs theorem.
- Author PDF for Nominal Algebra and the HSP Theorem
- Institutional record for Nominal Algebra and the HSP Theorem
- BibTeX for Nominal Algebra and the HSP Theorem @techreport{gabbay:nomahs-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra and the HSP Theorem}}, institution = {Heriot-Watt University}, number = {HW-MACS-TR-0057}, year = {2007}, month = nov, url = {https://research.tue.nl/en/publications/nominal-algebra-and-the-hsp-theorem/} }
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Publication type: Conference paper
A Formal Calculus for Informal Equality with Binding
WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation Lecture Notes in Computer Science vol. 4576 pp. 162–176 Springer
Nominal Algebra, a logic for equality reasoning with freshness and binding, based on nominal terms. See also applications Capture-avoiding substitution as a nominal algebra and One-and-a-halfth order logic in 2006, and the later journal version.
- Author PDF for A Formal Calculus for Informal Equality with Binding
- Publisher / DOI for A Formal Calculus for Informal Equality with Binding
- BibTeX for A Formal Calculus for Informal Equality with Binding @inproceedings{gabbay:forcie, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{A Formal Calculus for Informal Equality with Binding}}, booktitle = {WoLLIC'07: 14th Workshop on Logic, Language, Information and Computation}, series = {Lecture Notes in Computer Science}, volume = {4576}, pages = {162--176}, publisher = {Springer}, year = {2007}, month = jul, doi = {10.1007/978-3-540-73445-1_12}, url = {https://doi.org/10.1007/978-3-540-73445-1_12} }
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Publication type: Conference paper
a-logic with arrows
Proceedings of the 16th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2007), held at RDP 2007 pp. 47–62 Paris, France
Subsumed by the journal version. A previous paper on a-logic includes a predicate ‘atom’ related to ‘isvar&rsquo from PROLOG; to identify a term as a variable symbol. We give an overhauled proof-system, and a new denotational semantics, which use a directed notion of equality similar to a rewrite relation. We get improved ability to axiomatise logic and computation. See also the proceedings of the WFLP07 workshop.
- Author PDF for a-logic with arrows
- BibTeX for a-logic with arrows @inproceedings{gabbay:alwa, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic with arrows}}, booktitle = {Proceedings of the 16th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2007), held at RDP 2007}, pages = {47--62}, address = {Paris, France}, year = {2007}, month = jun, url = {https://www.gabbay.org.uk/papers/alwa.pdf} }
- Later version DOI for a-logic with arrows
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Publication type: Journal article
Fresh Logic: proof-theory and semantics for FM and nominal techniques
Journal of Applied Logic vol. 5, no. 2 pp. 356–387 Elsevier
The first proof-theory for the NEW quantifier, with sound and complete semantics. See also the subsequent LICS paper. The conference paper came into print three years earlier, though it was written later.
- Author PDF for Fresh Logic: proof-theory and semantics for FM and nominal techniques
- Publisher / DOI for Fresh Logic: proof-theory and semantics for FM and nominal techniques
- BibTeX for Fresh Logic: proof-theory and semantics for FM and nominal techniques @article{gabbay:frelog, author = {Murdoch J. Gabbay}, title = {{Fresh Logic: proof-theory and semantics for FM and nominal techniques}}, journal = {Journal of Applied Logic}, volume = {5}, number = {2}, pages = {356--387}, publisher = {Elsevier}, year = {2007}, month = jun, doi = {10.1016/j.jal.2005.10.012}, url = {https://doi.org/10.1016/j.jal.2005.10.012} }
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Publication type: Workshop abstract
Nominal Algebra
Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06) 3 pp. total Reykjavík, Iceland
We describe Nominal Algebra, an algebraic system which can directly encode systems-with-binding without using functions. See also Slides of the accompanying talk, and a paper.
- Author PDF for Nominal Algebra
- Institutional record for Nominal Algebra
- BibTeX for Nominal Algebra @inproceedings{gabbay:noma-nwpt, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra}}, booktitle = {Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06)}, address = {Reykjav{\'i}k, Iceland}, year = {2006}, month = oct, note = {3 pages}, url = {https://research.tue.nl/en/publications/nominal-algebra/} }
- Later version DOI for Nominal Algebra
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Publication type: Conference paper
One-and-a-halfth-order logic
Proceedings of the 8th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2006) pp. 189–200 ACM ISBN 1-59593-388-3
A logic for schematic first-order reasoning; unknown predicates are nominal unknowns in the sense of nominal terms and may occur under binders. We characterise derivability using nominal algebra and using sequents, prove equivalence, cut-elimination, and correctness. See the journal version.
- Author PDF for One-and-a-halfth-order logic
- Publisher / DOI for One-and-a-halfth-order logic
- BibTeX for One-and-a-halfth-order logic @inproceedings{gabbay:oneaah, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{One-and-a-halfth-order logic}}, booktitle = {Proceedings of the 8th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2006)}, pages = {189--200}, publisher = {ACM}, year = {2006}, month = jul, doi = {10.1145/1140335.1140359}, url = {https://doi.org/10.1145/1140335.1140359} }
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Publication type: Technical report
Nominal Algebra
Heriot-Watt University Technical Report no. HW-MACS-TR-0045 13 pp. total Edinburgh, UK
A detailed treatment of the nominal algebra presented in our abstract in NWPT06. Subsumed by a journal version.
- Author PDF for Nominal Algebra
- Institutional record for Nominal Algebra
- BibTeX for Nominal Algebra @techreport{gabbay:noma-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Nominal Algebra}}, institution = {Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0045}, address = {Edinburgh, UK}, year = {2006}, note = {13 pages}, url = {https://research.tue.nl/en/publications/nominal-algebra-2/} }
- Later version DOI for Nominal Algebra
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Publication type: Book chapter
Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
We Will Show Them: Essays in Honour of Dov Gabbay eds. S. Artemov, H. Barringer, A. d'Avila Garcez, Luis C. Lamb, J. Woods vol. 1 pp. 701–730 London: College Publications ISBN 1-904987-11-7 (pb); 1-904987-25-7 (hb)
We investigate Restart a natural deduction rule which reads “from A we may deduce B”. This powerful rule by Dov Gabbay in the 1980s adds many theorems to intuitionistic logic; we investigate them and consider generalisations. This is not a hoax.
- Author PDF for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- Publisher page for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- BibTeX for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning @incollection{gabbay:somfcg, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning}}, booktitle = {We Will Show Them: Essays in Honour of Dov Gabbay}, editor = {S. Artemov and H. Barringer and A. d'Avila Garcez and Luis C. Lamb and J. Woods}, volume = {1}, pages = {701--730}, publisher = {College Publications}, address = {London}, year = {2005}, month = oct, isbn = {1-904987-11-7 (pb); 1-904987-25-7 (hb)}, url = {https://www.collegepublications.co.uk/tributes/?00002} }
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Publication type: Book chapter
a-logic
We Will Show Them: Essays in Honour of Dov Gabbay eds. S. Artemov, H. Barringer, A. d'Avila Garcez, Luis C. Lamb, J. Woods vol. 1 pp. 731–764 London: College Publications ISBN 1-904987-11-7 (pb); 1-904987-25-7 (hb)
We extend first-order logic with a predicate expressing the intensional property of a term “this is a variable symbol”. We axiomatise substition, object-level binding, and thus exhibit a finite first-order axiomatisation of the lambda-calculus. See also a related development.
- Author PDF for a-logic
- Publisher page for a-logic
- BibTeX for a-logic @incollection{gabbay:alogic, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{a-logic}}, booktitle = {We Will Show Them: Essays in Honour of Dov Gabbay}, editor = {S. Artemov and H. Barringer and A. d'Avila Garcez and Luis C. Lamb and J. Woods}, volume = {1}, pages = {731--764}, publisher = {College Publications}, address = {London}, year = {2005}, month = oct, isbn = {1-904987-11-7 (pb); 1-904987-25-7 (hb)}, url = {https://www.collegepublications.co.uk/tributes/?00002} }
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Publication type: Conference paper
A Sequent Calculus for Nominal Logic
Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004) pp. 139–148 IEEE Computer Society
A sequent calculus for Nominal Logic. We consider notions of Logic Programming in it and give a translation of Miller and Tiu’s FOLN.
- Author PDF for A Sequent Calculus for Nominal Logic
- Publisher / DOI for A Sequent Calculus for Nominal Logic
- BibTeX for A Sequent Calculus for Nominal Logic @inproceedings{gabbay:seqcnl, author = {Murdoch J. Gabbay and James Cheney}, title = {{A Sequent Calculus for Nominal Logic}}, booktitle = {Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004)}, pages = {139--148}, publisher = {IEEE Computer Society}, year = {2004}, month = jul, doi = {10.1109/LICS.2004.6}, url = {https://doi.org/10.1109/LICS.2004.6} }
Nominal sets and foundations 23 publications
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Publication type: Preprint
Nominal techniques as an Agda library
- Author PDF for Nominal techniques as an Agda library
- arXiv for Nominal techniques as an Agda library
- BibTeX for Nominal techniques as an Agda library @misc{gabbay-melkonian:nominal-agda-library, author = {Murdoch J. Gabbay and Orestis Melkonian}, title = {{Nominal techniques as an Agda library}}, howpublished = {arXiv preprint arXiv:2603.03968}, year = {2026}, month = mar, doi = {10.48550/arXiv.2603.03968}, eprint = {2603.03968}, archivePrefix = {arXiv}, primaryClass = {cs.PL}, url = {https://doi.org/10.48550/arXiv.2603.03968} }
- Code for Nominal techniques as an Agda library
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Publication type: Journal article
Equivariant ZFA and the foundations of nominal techniques
Journal of Logic and Computation vol. 30, no. 2 pp. 525–548 ISSN 0955-792X online
A new take on the foundations of nominal techniques.
- Author PDF for Equivariant ZFA and the foundations of nominal techniques
- Publisher / DOI for Equivariant ZFA and the foundations of nominal techniques
- BibTeX for Equivariant ZFA and the foundations of nominal techniques @article{gabbay:equzfn, author = {Murdoch J. Gabbay}, title = {{Equivariant ZFA and the foundations of nominal techniques}}, journal = {Journal of Logic and Computation}, volume = {30}, number = {2}, pages = {525--548}, year = {2020}, month = mar, doi = {10.1093/logcom/exz015}, url = {https://doi.org/10.1093/logcom/exz015} }
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Publication type: Journal article
The language of Stratified Sets is confluent and strongly normalising
Logical Methods in Computer Science vol. 14, no. 2 article 12
Quine’s stratification condition viewed from the perspective of the theory of rewriting.
- Author PDF for The language of Stratified Sets is confluent and strongly normalising
- Publisher / DOI for The language of Stratified Sets is confluent and strongly normalising
- BibTeX for The language of Stratified Sets is confluent and strongly normalising @article{gabbay:lanssc, author = {Murdoch J. Gabbay}, title = {{The language of Stratified Sets is confluent and strongly normalising}}, journal = {Logical Methods in Computer Science}, volume = {14}, number = {2}, eid = {12}, year = {2018}, month = may, note = {Article 12}, doi = {10.23638/LMCS-14(2:12)2018}, url = {https://doi.org/10.23638/LMCS-14(2:12)2018} }
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Publication type: Conference paper
Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
24th EACSL Annual Conference on Computer Science Logic (CSL 2015) ed. Kreutzer, Stephan Leibniz International Proceedings in Informatics (LIPIcs) vol. 41 pp. 374–389 Dagstuhl, Germany: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik ISBN 978-3-939897-90-3 ISSN 1868-8969
Split “νa.t” as “〈a t a〉”. If that looks simple, try “〈a〈b ab a〉b〉” or “〈a〈b abb〉”.
- Author PDF for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
- Publisher / DOI for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes
- BibTeX for Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes @inproceedings{gabbay:leatnn, author = {Murdoch J. Gabbay and Dan R. Ghica and Daniela Petrisan}, title = {{Leaving the Nest: Nominal Techniques for Variables with Interleaving Scopes}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, volume = {41}, pages = {374--389}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}, year = {2015}, month = sep, doi = {10.4230/LIPIcs.CSL.2015.374}, url = {https://doi.org/10.4230/LIPIcs.CSL.2015.374} }
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Publication type: Book chapter
Nominal terms and nominal logics: from foundations to meta-mathematics
Handbook of Philosophical Logic vol. 17 pp. 79–178 Springer
A survey and update chapter describing a journey from (permissive-)nominal sets, via nominal terms, rewriting and algebra, to permissive-nominal logic and a finite first-order nominal axiomatisation of arithmetic. See also Foundations of nominal techniques.
- Author PDF for Nominal terms and nominal logics: from foundations to meta-mathematics
- Publisher / DOI for Nominal terms and nominal logics: from foundations to meta-mathematics
- BibTeX for Nominal terms and nominal logics: from foundations to meta-mathematics @incollection{gabbay:nomtnl, author = {Murdoch J. Gabbay}, title = {{Nominal terms and nominal logics: from foundations to meta-mathematics}}, booktitle = {Handbook of Philosophical Logic}, volume = {17}, pages = {79--178}, publisher = {Springer}, year = {2013}, month = jan, doi = {10.1007/978-94-007-6600-6_2}, url = {https://doi.org/10.1007/978-94-007-6600-6_2} }
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Publication type: Journal article
Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
Mathematical Structures in Computer Science vol. 21, no. 5 pp. 997–1033 Cambridge University Press
A nominal denotational semantics for meta-variables. Nominal techniques have already given semantics to variables; in this paper we show how they can do the same for meta-variables. The version on this webpage includes corrections.
- Author PDF for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- Publisher / DOI for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- BibTeX for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables @article{gabbay:twolns, author = {Murdoch J. Gabbay}, title = {{Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables}}, journal = {Mathematical Structures in Computer Science}, volume = {21}, number = {5}, pages = {997--1033}, publisher = {Cambridge University Press}, year = {2011}, month = oct, doi = {10.1017/S0960129511000272}, url = {https://doi.org/10.1017/S0960129511000272} }
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Publication type: Journal article
Foundations of nominal techniques: logic and semantics of variables in abstract syntax
Bulletin of Symbolic Logic vol. 17, no. 2 pp. 161–229
A survey and update paper describing nominal techniques, with specific application to nominal abstract syntax. A sequel to this paper is Nominal terms and nominal logics.
- Author PDF for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- Publisher / DOI for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- BibTeX for Foundations of nominal techniques: logic and semantics of variables in abstract syntax @article{gabbay:fountl, author = {Murdoch J. Gabbay}, title = {{Foundations of nominal techniques: logic and semantics of variables in abstract syntax}}, journal = {Bulletin of Symbolic Logic}, volume = {17}, number = {2}, pages = {161--229}, year = {2011}, month = jun, doi = {10.2178/bsl/1305810911}, url = {https://doi.org/10.2178/bsl/1305810911} }
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Publication type: Conference paper
Freshness and name-restriction in sets of traces with names
Foundations of software science and computation structures, 14th International Conference (FOSSACS 2011) Lecture Notes in Computer Science vol. 6604 pp. 365–380 Springer
A concrete sets model of name-generation/name-restriction using a relative of Gabbay-Pitts atoms-abstraction. The technical innovation is the notion of a positive subset of a nominal set.
- Author PDF for Freshness and name-restriction in sets of traces with names
- Publisher / DOI for Freshness and name-restriction in sets of traces with names
- BibTeX for Freshness and name-restriction in sets of traces with names @inproceedings{gabbay:frenrs, author = {Murdoch J. Gabbay and Vincenzo Ciancia}, title = {{Freshness and name-restriction in sets of traces with names}}, booktitle = {Foundations of software science and computation structures, 14th International Conference (FOSSACS 2011)}, series = {Lecture Notes in Computer Science}, volume = {6604}, pages = {365--380}, publisher = {Springer}, year = {2011}, month = mar, doi = {10.1007/978-3-642-19805-2_25}, url = {https://doi.org/10.1007/978-3-642-19805-2_25} }
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Publication type: Technical report
An observation on support and freshness in nominal sets
School of Mathematical and Computer Sciences, Heriot-Watt University Technical Report no. HW-MACS-TR-0077 14 pp. total Edinburgh, UK
There are two natural ways of excluding an atom a in nominal techniques: we can consider the sets X such that a is fresh for X, or we can consider the sets X such that a is fresh for every x ∈ X. We show these lead to isomorphic categories. This can be read as saying that a fresh atom in the object-level is categorically isomorphic to a fresh atom in the meta-level.
- Author PDF for An observation on support and freshness in nominal sets
- BibTeX for An observation on support and freshness in nominal sets @techreport{gabbay:obssfn-tr, author = {Murdoch J. Gabbay}, title = {{An observation on support and freshness in nominal sets}}, institution = {School of Mathematical and Computer Sciences, Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0077}, address = {Edinburgh, UK}, year = {2010}, month = may, note = {14 pages}, url = {https://www.gabbay.org.uk/papers/obssfn-tr.pdf} }
- Catalogue record for An observation on support and freshness in nominal sets
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Publication type: Conference paper
Semantic nominal terms
Second International Workshop on Theory and Applications of Abstraction, Substitution, and Naming (TAASN 2009), associated with ETAPS 2009 5 pp. total York, UK
We reconcile nominal atoms-abstraction with nominal terms. See the journal paper on two-level nominal sets.
- Author PDF for Semantic nominal terms
- BibTeX for Semantic nominal terms @inproceedings{gabbay:semnt, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Semantic nominal terms}}, booktitle = {Second International Workshop on Theory and Applications of Abstraction, Substitution, and Naming (TAASN 2009), associated with ETAPS 2009}, address = {York, UK}, year = {2009}, month = mar, note = {5 pages}, url = {https://www.gabbay.org.uk/papers/semnt-ea.pdf} }
- Later version DOI for Semantic nominal terms
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Publication type: Conference paper
Nominal renaming sets
Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008) pp. 158–173 Springer
Nominal renaming sets are like nominal sets, but they are based on a finitely-supported renaming action — a renaming is a function from atoms to atoms, not necessarily injective. This provides a convenient sets-based semantics with which to justify Higher-Order Abstract Syntax.
- Author PDF for Nominal renaming sets
- Publisher / DOI for Nominal renaming sets
- BibTeX for Nominal renaming sets @inproceedings{gabbay:nomrs, author = {Murdoch J. Gabbay and Martin Hofmann}, title = {{Nominal renaming sets}}, booktitle = {Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008)}, pages = {158--173}, publisher = {Springer}, year = {2008}, month = nov, doi = {10.1007/978-3-540-89439-1_11}, url = {https://doi.org/10.1007/978-3-540-89439-1_11} }
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Publication type: Workshop abstract
Arbitrary Objects in Mathematics and Semantics
Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics 2 pp. total Paris, France
A photo of Michael delivering the talk.
- Author PDF for Arbitrary Objects in Mathematics and Semantics
- BibTeX for Arbitrary Objects in Mathematics and Semantics @inproceedings{gabbay:arboms, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Arbitrary Objects in Mathematics and Semantics}}, booktitle = {Paris Arch{\'e} workshop on abstract objects in semantics and the philosophy of mathematics}, address = {Paris, France}, year = {2008}, month = feb, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/arboms.pdf} }
- Catalogue record for Arbitrary Objects in Mathematics and Semantics
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Publication type: Technical report
Nominal renaming sets
Heriot-Watt University Technical Report no. HW-MACS-TR-0058 35 pp. total Edinburgh, UK
Subsumed by my LPAR08 paper with Martin Hofmann. Nominal sets based on a renaming action (possibly non-injective functions on atoms) instead of the more usual permutation action.
- Author PDF for Nominal renaming sets
- BibTeX for Nominal renaming sets @techreport{gabbay:nomrs-tr, author = {Murdoch J. Gabbay}, title = {{Nominal renaming sets}}, institution = {Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0058}, address = {Edinburgh, UK}, year = {2007}, month = dec, note = {35 pages}, url = {https://www.gabbay.org.uk/papers/nomrs-tr.pdf} }
- Later version DOI for Nominal renaming sets
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Publication type: Book chapter
Curry-style types for nominal terms
Types for Proofs and Programs (TYPES 06) Lecture Notes in Computer Science vol. 4502 pp. 125–139 Springer
Nominal terms are a representation of syntax-with-binding in the presence of explicit meta-variables and alpha-renaming. Making this consistent with a functional type system is non-trivial because the same meta-variable may appear in different contexts and under different binders, so when that meta-variable is instantiated the same variable name may be injected into different contexts of abstractions.
- Author PDF for Curry-style types for nominal terms
- Publisher / DOI for Curry-style types for nominal terms
- BibTeX for Curry-style types for nominal terms @incollection{gabbay:curstn, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Curry-style types for nominal terms}}, booktitle = {Types for Proofs and Programs (TYPES 06)}, series = {Lecture Notes in Computer Science}, volume = {4502}, pages = {125--139}, publisher = {Springer}, year = {2007}, month = sep, doi = {10.1007/978-3-540-74464-1_9}, url = {https://doi.org/10.1007/978-3-540-74464-1_9} }
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Publication type: Journal article
A General Mathematics of Names
Information and Computation vol. 205, no. 7 pp. 982–1011 Academic Press
Nominal sets as introduced here have an atoms-abstraction operation generalising alpha-equivalence. In this paper we consider infinite simultaneous alpha-equivalence. The trick is to interpret small as well-orderable.
- Author PDF for A General Mathematics of Names
- Publisher / DOI for A General Mathematics of Names
- BibTeX for A General Mathematics of Names @article{gabbay:genmn, author = {Murdoch J. Gabbay}, title = {{A General Mathematics of Names}}, journal = {Information and Computation}, volume = {205}, number = {7}, pages = {982--1011}, publisher = {Academic Press}, year = {2007}, month = jul, doi = {10.1016/j.ic.2006.10.010}, url = {https://doi.org/10.1016/j.ic.2006.10.010} }
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Publication type: Book chapter
Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
We Will Show Them: Essays in Honour of Dov Gabbay eds. S. Artemov, H. Barringer, A. d'Avila Garcez, Luis C. Lamb, J. Woods vol. 1 pp. 701–730 London: College Publications ISBN 1-904987-11-7 (pb); 1-904987-25-7 (hb)
We investigate Restart a natural deduction rule which reads “from A we may deduce B”. This powerful rule by Dov Gabbay in the 1980s adds many theorems to intuitionistic logic; we investigate them and consider generalisations. This is not a hoax.
- Author PDF for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- Publisher page for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning
- BibTeX for Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning @incollection{gabbay:somfcg, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{Some Formal Considerations on Gabbay's Restart Rule in Natural Deduction and Goal-Directed Reasoning}}, booktitle = {We Will Show Them: Essays in Honour of Dov Gabbay}, editor = {S. Artemov and H. Barringer and A. d'Avila Garcez and Luis C. Lamb and J. Woods}, volume = {1}, pages = {701--730}, publisher = {College Publications}, address = {London}, year = {2005}, month = oct, isbn = {1-904987-11-7 (pb); 1-904987-25-7 (hb)}, url = {https://www.collegepublications.co.uk/tributes/?00002} }
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Publication type: Conference paper
A NEW calculus of contexts
Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05) eds. Pedro Barahona, Amy P. Felty pp. 94–105 Lisbon, Portugal: Association for Computing Machinery ISBN 1-59593-090-6
Subsumed by a later journal paper. A lambda-calculus with context substitution (which does not avoid capture) alongside ordinary beta-reduction. It can encode dynamic binding, objects, and interesting properties associated to partial evaluation. Meta-properties include an applicative characterisation of contextual equivalence.
- Author PDF for A NEW calculus of contexts
- Publisher / DOI for A NEW calculus of contexts
- BibTeX for A NEW calculus of contexts @inproceedings{gabbay:newcc, author = {Murdoch J. Gabbay}, title = {{A NEW calculus of contexts}}, booktitle = {Proceedings of the 7th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP '05)}, editor = {Pedro Barahona and Amy P. Felty}, pages = {94--105}, publisher = {Association for Computing Machinery}, address = {Lisbon, Portugal}, year = {2005}, month = jul, isbn = {1-59593-090-6}, doi = {10.1145/1069774.1069783}, url = {https://doi.org/10.1145/1069774.1069783} }
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Publication type: Conference paper
Automating Fraenkel-Mostowski Syntax
Track B Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002) eds. Víctor A. Carreño, César A. Muñoz, Sofiène Tahar NASA Conference Publication vol. NASA/CP-2002-211736 pp. 60–70 Hampton, Virginia, USA: National Aeronautics and Space Administration, Langley Research Center
- Author PDF for Automating Fraenkel-Mostowski Syntax
- NASA proceedings PDF for Automating Fraenkel-Mostowski Syntax
- BibTeX for Automating Fraenkel-Mostowski Syntax @inproceedings{gabbay:autfms, author = {Murdoch J. Gabbay}, title = {{Automating Fraenkel-Mostowski Syntax}}, booktitle = {Track B Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002)}, editor = {V{\'i}ctor A. Carre{\~n}o and C{\'e}sar A. Mu{\~n}oz and Sofi{\`e}ne Tahar}, series = {NASA Conference Publication}, volume = {NASA/CP-2002-211736}, pages = {60--70}, publisher = {National Aeronautics and Space Administration, Langley Research Center}, address = {Hampton, Virginia, USA}, year = {2002}, month = aug, url = {https://ntrs.nasa.gov/api/citations/20020063594/downloads/20020063594.pdf} }
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Publication type: Journal article
A New Approach to Abstract Syntax with Variable Binding
Formal Aspects of Computing vol. 13, no. 3–5 pp. 341–363
The paper that started it all, at least for me. We introduce nominal sets, atoms-abstraction, the NEW quantifier, and nominal abstract syntax (inductive datatypes of syntax-with-binding) and their inductive reasoning principles. See also the conference version in LICS99.
- Author PDF for A New Approach to Abstract Syntax with Variable Binding
- Publisher / DOI for A New Approach to Abstract Syntax with Variable Binding
- BibTeX for A New Approach to Abstract Syntax with Variable Binding @article{gabbay:newaas-jv, author = {Murdoch J. Gabbay and Andrew M. Pitts}, title = {{A New Approach to Abstract Syntax with Variable Binding}}, journal = {Formal Aspects of Computing}, volume = {13}, number = {3--5}, pages = {341--363}, year = {2002}, month = jul, doi = {10.1007/s001650200016}, url = {https://doi.org/10.1007/s001650200016} }
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Publication type: Conference paper
FM-HOL, a higher-order theory of names
Informal Proceedings of the Workshop on Thirty Five Years of Automath ed. Fairouz Kamareddine Edinburgh, Scotland: Heriot-Watt University
A generalisation of Fraenkel-Mostowski techniques to deal with infinite support and infinite atoms-abstraction. Largely but not completely superseded by a general mathematics of names.
- Author PDF for FM-HOL, a higher-order theory of names
- Workshop PDF for FM-HOL, a higher-order theory of names
- BibTeX for FM-HOL, a higher-order theory of names @inproceedings{gabbay:fmhotn, author = {Murdoch J. Gabbay}, title = {{FM-HOL, a higher-order theory of names}}, booktitle = {Informal Proceedings of the Workshop on Thirty Five Years of Automath}, editor = {Fairouz Kamareddine}, publisher = {Heriot-Watt University}, address = {Edinburgh, Scotland}, year = {2002}, month = apr, url = {https://www.macs.hw.ac.uk/~fairouz/forest/events/automath2002/informal-proceedings/ACC/gabbay.pdf} }
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Publication type: Workshop abstract
Nominal Sets, Equivariance Reasoning, and Variable Binding
Second Workshop on Meta-Logic (MLG 2002) 2 pp. total Kinosaki, Japan
A prose account of my talk at MLG02 in Kinosaki, Japan.
- Author PDF for Nominal Sets, Equivariance Reasoning, and Variable Binding
- BibTeX for Nominal Sets, Equivariance Reasoning, and Variable Binding @inproceedings{gabbay:nomser, author = {Murdoch J. Gabbay}, title = {{Nominal Sets, Equivariance Reasoning, and Variable Binding}}, booktitle = {Second Workshop on Meta-Logic (MLG 2002)}, address = {Kinosaki, Japan}, year = {2002}, note = {2 pages}, url = {https://www.gabbay.org.uk/papers/nomser.pdf} }
- Catalogue record for Nominal Sets, Equivariance Reasoning, and Variable Binding
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Publication type: Doctoral thesis
A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
University of Cambridge Cambridge, UK
See also the survey paper on foundations on nominal techniques.
- Author PDF for A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language
- BibTeX for A Theory of Inductive Definitions with α-Equivalence: Semantics, Implementation, Programming Language @phdthesis{gabbay:thesis, author = {Murdoch J. Gabbay}, title = {{A Theory of Inductive Definitions with {$\alpha$}-Equivalence: Semantics, Implementation, Programming Language}}, school = {University of Cambridge}, address = {Cambridge, UK}, year = {2001}, month = mar, url = {https://www.gabbay.org.uk/papers/thesis.pdf} }
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Publication type: Conference paper
A New Approach to Abstract Syntax Involving Binders
Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS 1999) pp. 214–224 IEEE Computer Society Press
My first conference paper. See also the journal version.
- Author PDF for A New Approach to Abstract Syntax Involving Binders
- Publisher / DOI for A New Approach to Abstract Syntax Involving Binders
- BibTeX for A New Approach to Abstract Syntax Involving Binders @inproceedings{gabbay:newaas, author = {Murdoch J. Gabbay and Andrew M. Pitts}, title = {{A New Approach to Abstract Syntax Involving Binders}}, booktitle = {Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS 1999)}, pages = {214--224}, publisher = {IEEE Computer Society Press}, year = {1999}, month = jul, doi = {10.1109/LICS.1999.782617}, url = {https://doi.org/10.1109/LICS.1999.782617} }
Programming languages and processes 8 publications
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Publication type: Preprint
Nominal techniques as an Agda library
- Author PDF for Nominal techniques as an Agda library
- arXiv for Nominal techniques as an Agda library
- BibTeX for Nominal techniques as an Agda library @misc{gabbay-melkonian:nominal-agda-library, author = {Murdoch J. Gabbay and Orestis Melkonian}, title = {{Nominal techniques as an Agda library}}, howpublished = {arXiv preprint arXiv:2603.03968}, year = {2026}, month = mar, doi = {10.48550/arXiv.2603.03968}, eprint = {2603.03968}, archivePrefix = {arXiv}, primaryClass = {cs.PL}, url = {https://doi.org/10.48550/arXiv.2603.03968} }
- Code for Nominal techniques as an Agda library
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Publication type: Conference paper
UTxO- vs account-based smart contract blockchain programming paradigms
Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020) eds. Tiziana Margaria, Bernhard Steffen Lecture Notes in Computer Science vol. 12478 pp. 73–88 Springer
We implement two versions of a simple but illustrative smart contract: one in Solidity on the Ethereum blockchain platform, and one in Plutus on the Cardano platform, with annotated code excerpts and with source code attached. We prove some simple but novel results about observational equivalence and conclude with a wide-ranging and detailed discussion in the light of the examples, mathematical model, and mathematical results so far.
- Author PDF for UTxO- vs account-based smart contract blockchain programming paradigms
- Publisher / DOI for UTxO- vs account-based smart contract blockchain programming paradigms
- BibTeX for UTxO- vs account-based smart contract blockchain programming paradigms @inproceedings{gabbay:utxabs, author = {Lars Br{\"u}njes and Murdoch J. Gabbay}, title = {{UTxO- vs account-based smart contract blockchain programming paradigms}}, booktitle = {Proceedings of the 9th International Symposium On Leveraging Applications of Formal Methods, Verification and Validation (ISOLA 2020)}, editor = {Tiziana Margaria and Bernhard Steffen}, series = {Lecture Notes in Computer Science}, volume = {12478}, pages = {73--88}, publisher = {Springer}, year = {2020}, month = oct, doi = {10.1007/978-3-030-61467-6_6}, eprint = {2003.14271}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1007/978-3-030-61467-6_6} }
- arXiv preprint for UTxO- vs account-based smart contract blockchain programming paradigms
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Publication type: Journal article
Nominal SOS
Electronic Notes in Theoretical Computer Science vol. 286 Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012) pp. 103–116 Elsevier ISSN 1571-0661
- Publisher / DOI for Nominal SOS
- BibTeX for Nominal SOS @article{cimini-mousavi-reniers-gabbay:nomsos, author = {Matteo Cimini and Mohammad Reza Mousavi and Michel A. Reniers and Murdoch J. Gabbay}, title = {{Nominal SOS}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {286}, pages = {103--116}, publisher = {Elsevier}, year = {2012}, month = sep, issn = {1571-0661}, note = {Special issue: Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012)}, doi = {10.1016/j.entcs.2012.08.008}, url = {https://doi.org/10.1016/j.entcs.2012.08.008} }
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Publication type: Workshop abstract
Nominal SOS
Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06) 2 pp. total Reykjavík, Iceland
We describe Nominal Structured Operational Semantics, a framework for specifying operational semantics within which proofs of congruence are carried out ‘once and for all’.
- Author PDF for Nominal SOS
- Workshop PDF for Nominal SOS
- BibTeX for Nominal SOS @inproceedings{gabbay:nomsos-nwpt, author = {MohammadReza Mousavi and Michel A. Reniers and Murdoch J. Gabbay}, title = {{Nominal SOS}}, booktitle = {Proceedings of the 18th Nordic Workshop on Programming Theory (NWPT'06)}, address = {Reykjav{\'i}k, Iceland}, year = {2006}, month = oct, note = {2 pages}, url = {https://staff.ru.is/luca/NWPT06/PAPERS/paper1.pdf} }
- Later version DOI for Nominal SOS
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Publication type: Conference paper
SOS for Higher Order Processes (Extended Abstract)
Proceedings of the 16th International Conference on Concurrency Theory (CONCUR'05) eds. Abadi, Martin; de Alfaro, Luca Lecture Notes in Computer Science vol. 3653 pp. 308–322 Springer
Promoted PANTH is a Structural Operational Semantics framework extends tyft/tyxt, for which a notion of higher-order bisimulation is a congruence. (Slides by MohammadReza Mousavi.)
- Author PDF for SOS for Higher Order Processes (Extended Abstract)
- Publisher / DOI for SOS for Higher Order Processes (Extended Abstract)
- BibTeX for SOS for Higher Order Processes (Extended Abstract) @inproceedings{gabbay:soshop, author = {MohammadReza Mousavi and Murdoch J. Gabbay and Michel A. Reniers}, title = {{SOS for Higher Order Processes (Extended Abstract)}}, booktitle = {Proceedings of the 16th International Conference on Concurrency Theory (CONCUR'05)}, series = {Lecture Notes in Computer Science}, volume = {3653}, pages = {308--322}, publisher = {Springer}, year = {2005}, month = sep, doi = {10.1007/11539452_25}, url = {https://doi.org/10.1007/11539452_25} }
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Publication type: Book chapter
The π-Calculus in FM
Thirty-five years of Automating Mathematics ed. Fairouz D. Kamareddine Applied Logic Series vol. 28 pp. 247–269 Dordrecht: Kluwer Academic Publishers edition 1 ISBN 978-1-4020-1656-1 eISBN 978-94-017-0253-9
We discuss how to use nominal techniques to model name-generation, specifically in a process calculus, though the techniques use are probably more generally applicable. (Electronic versions differ in minor corrections from the printed version.)
- Author PDF for The π-Calculus in FM
- Publisher / DOI for The π-Calculus in FM
- BibTeX for The π-Calculus in FM @incollection{gabbay:picfm, author = {Murdoch J. Gabbay}, title = {{The {$\pi$}-Calculus in FM}}, booktitle = {Thirty-five years of Automating Mathematics}, editor = {Fairouz D. Kamareddine}, series = {Applied Logic Series}, volume = {28}, pages = {247--269}, publisher = {Kluwer Academic Publishers}, address = {Dordrecht}, edition = {1}, year = {2003}, month = nov, isbn = {978-1-4020-1656-1}, note = {Electronic ISBN 978-94-017-0253-9}, doi = {10.1007/978-94-017-0253-9_10}, url = {https://doi.org/10.1007/978-94-017-0253-9_10} }
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Publication type: Conference paper
FreshML: Programming with Binders Made Simple
SIGPLAN Notices vol. 38 pp. 263–274 ACM Press
ML enriched with Gabbay-Pitts atoms-abstraction and nominal name-generation and pattern-matching.
- Author PDF for FreshML: Programming with Binders Made Simple
- Publisher / DOI for FreshML: Programming with Binders Made Simple
- BibTeX for FreshML: Programming with Binders Made Simple @inproceedings{gabbay:frepbm, author = {Mark R. Shinwell and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{FreshML: Programming with Binders Made Simple}}, booktitle = {Proceedings of the 8th ACM SIGPLAN International Conference on Functional Programming (ICFP 2003)}, journal = {SIGPLAN Notices}, volume = {38}, pages = {263--274}, publisher = {ACM Press}, year = {2003}, month = aug, doi = {10.1145/944705.944729}, url = {https://doi.org/10.1145/944705.944729} }
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Publication type: Conference paper
A Metalanguage for Programming with Bound Names Modulo Renaming
Proceedings of the 5th international conference on the mathematics of program construction (MPC 2000) Lecture Notes in Computer Science vol. 1837 pp. 230–255 Springer
- Author PDF for A Metalanguage for Programming with Bound Names Modulo Renaming
- Publisher / DOI for A Metalanguage for Programming with Bound Names Modulo Renaming
- BibTeX for A Metalanguage for Programming with Bound Names Modulo Renaming @inproceedings{gabbay:metpbn, author = {Andrew M. Pitts and Murdoch J. Gabbay}, title = {{A Metalanguage for Programming with Bound Names Modulo Renaming}}, booktitle = {Proceedings of the 5th international conference on the mathematics of program construction (MPC 2000)}, series = {Lecture Notes in Computer Science}, volume = {1837}, pages = {230--255}, publisher = {Springer}, year = {2000}, month = dec, doi = {10.1007/10722010_15}, url = {https://doi.org/10.1007/10722010_15} }
Rewriting and unification 22 publications
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Publication type: Conference paper
Equational Reasoning in Languages with Binders via Permutation Fixed-Points
11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026) ed. Frank Pfenning Leibniz International Proceedings in Informatics (LIPIcs) vol. 378 pp. 10:1–10:23 article 10 Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik ISBN 978-3-95977-433-8 ISSN 1868-8969
- Author PDF for Equational Reasoning in Languages with Binders via Permutation Fixed-Points
- Publisher / DOI for Equational Reasoning in Languages with Binders via Permutation Fixed-Points
- BibTeX for Equational Reasoning in Languages with Binders via Permutation Fixed-Points @inproceedings{caires-santos-fernandez-gabbay-nantes-sobrinho:permutation-fixed-points, author = {Ali K. Caires-Santos and Maribel Fern{\'a}ndez and Murdoch James Gabbay and Daniele Nantes-Sobrinho}, title = {{Equational Reasoning in Languages with Binders via Permutation Fixed-Points}}, booktitle = {11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026)}, editor = {Frank Pfenning}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, volume = {378}, pages = {10:1--10:23}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, year = {2026}, month = jul, isbn = {978-3-95977-433-8}, issn = {1868-8969}, doi = {10.4230/LIPIcs.FSCD.2026.10}, url = {https://doi.org/10.4230/LIPIcs.FSCD.2026.10} }
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Publication type: Journal article
The language of Stratified Sets is confluent and strongly normalising
Logical Methods in Computer Science vol. 14, no. 2 article 12
Quine’s stratification condition viewed from the perspective of the theory of rewriting.
- Author PDF for The language of Stratified Sets is confluent and strongly normalising
- Publisher / DOI for The language of Stratified Sets is confluent and strongly normalising
- BibTeX for The language of Stratified Sets is confluent and strongly normalising @article{gabbay:lanssc, author = {Murdoch J. Gabbay}, title = {{The language of Stratified Sets is confluent and strongly normalising}}, journal = {Logical Methods in Computer Science}, volume = {14}, number = {2}, eid = {12}, year = {2018}, month = may, note = {Article 12}, doi = {10.23638/LMCS-14(2:12)2018}, url = {https://doi.org/10.23638/LMCS-14(2:12)2018} }
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Publication type: Journal article
Checking Overlaps of Nominal Rewriting Rules
Electronic Notes in Theoretical Computer Science vol. 323 pp. 39–56 Elsevier B.V. ISSN 1571-0661
- Repository PDF for Checking Overlaps of Nominal Rewriting Rules
- Publisher / DOI for Checking Overlaps of Nominal Rewriting Rules
- BibTeX for Checking Overlaps of Nominal Rewriting Rules @article{ayalarincon:cheonr, author = {Mauricio Ayala-Rinc{\'o}n and Maribel Fern{\'a}ndez and Murdoch James Gabbay and Ana Cristina Rocha-Oliveira}, title = {{Checking Overlaps of Nominal Rewriting Rules}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {323}, pages = {39--56}, publisher = {Elsevier B.V.}, year = {2016}, month = jul, doi = {10.1016/j.entcs.2016.06.004}, url = {https://doi.org/10.1016/j.entcs.2016.06.004} }
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Publication type: Journal article
Meta-variables as infinite lists in nominal terms unification and rewriting
Logic Journal of the IGPL vol. 20, no. 6 pp. 967–1000 Oxford University Press
Nominal terms unknowns modelled concretely as infinite lists of distinct atoms. Theorems simplify and many even become immediate corollaries of equivariance.
- Author PDF for Meta-variables as infinite lists in nominal terms unification and rewriting
- Publisher / DOI for Meta-variables as infinite lists in nominal terms unification and rewriting
- BibTeX for Meta-variables as infinite lists in nominal terms unification and rewriting @article{gabbay:metvil, author = {Murdoch J. Gabbay}, title = {{Meta-variables as infinite lists in nominal terms unification and rewriting}}, journal = {Logic Journal of the IGPL}, volume = {20}, number = {6}, pages = {967--1000}, publisher = {Oxford University Press}, year = {2012}, month = dec, doi = {10.1093/jigpal/jzs005}, url = {https://doi.org/10.1093/jigpal/jzs005} }
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Publication type: Conference paper
Principal Types for Nominal Theories
Proceedings of the 18th International Symposium on Fundamentals of Computation Theory (FCT 2011) eds. Olaf Owe, Martin Steffen, Jan Arne Telle Lecture Notes in Computer Science vol. 6914 pp. 160–172 Berlin, Heidelberg: Springer ISBN 978-3-642-22952-7 eISBN 978-3-642-22953-4
Rank 1 polymorphic types for nominal rewriting and algebra. So you want to type rules like beta- or eta-conversion over nominal terms? This paper describes one way to do it.
- Author PDF for Principal Types for Nominal Theories
- Publisher / DOI for Principal Types for Nominal Theories
- BibTeX for Principal Types for Nominal Theories @inproceedings{gabbay:pritnt, author = {Elliot Fairweather and Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Principal Types for Nominal Theories}}, booktitle = {Proceedings of the 18th International Symposium on Fundamentals of Computation Theory (FCT 2011)}, editor = {Olaf Owe and Martin Steffen and Jan Arne Telle}, series = {Lecture Notes in Computer Science}, volume = {6914}, pages = {160--172}, publisher = {Springer}, address = {Berlin, Heidelberg}, year = {2011}, month = aug, isbn = {978-3-642-22952-7}, note = {Electronic ISBN 978-3-642-22953-4}, doi = {10.1007/978-3-642-22953-4_14}, url = {https://doi.org/10.1007/978-3-642-22953-4_14} }
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Publication type: Journal article
Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
Logic Journal of the IGPL vol. 18, no. 6 pp. 769–822
You cannot quotient nominal terms by alpha-equivalence, but you can quotient permissive nominal terms. This journal paper extends the conference version and technical report and includes some elements from another conference paper. My version includes minor corrections to the published version.
- Author PDF for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
- Publisher / DOI for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)
- BibTeX for Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version) @article{gabbay:perntu-jv, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification: an infinite, co-infinite approach to nominal techniques (journal version)}}, journal = {Logic Journal of the IGPL}, volume = {18}, number = {6}, pages = {769--822}, year = {2010}, month = dec, doi = {10.1093/jigpal/jzq006}, url = {https://doi.org/10.1093/jigpal/jzq006} }
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Publication type: Conference paper
Closed nominal rewriting and efficiently computable nominal algebra equality
Electronic Proceedings in Theoretical Computer Science vol. 34 pp. 37–51
We relate nominal algebra and rewriting, with concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality, which includes specifications of lambda-calculus and first-order logic.
- Author PDF for Closed nominal rewriting and efficiently computable nominal algebra equality
- Publisher / DOI for Closed nominal rewriting and efficiently computable nominal algebra equality
- BibTeX for Closed nominal rewriting and efficiently computable nominal algebra equality @inproceedings{gabbay:clonre, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Closed nominal rewriting and efficiently computable nominal algebra equality}}, booktitle = {Electronic Proceedings in Theoretical Computer Science}, volume = {34}, pages = {37--51}, year = {2010}, month = sep, doi = {10.4204/EPTCS.34.5}, url = {https://doi.org/10.4204/EPTCS.34.5} }
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Publication type: Conference paper
Permissive Nominal Terms and their Unification
Atti del Convegno CILC 09: 24-esimo Convegno Italiano di Logica Computazionale eds. Marco Gavanelli, Fabrizio Riguzzi pp. 70–84 Ferrara, Italy: GULP (Italian Association for Logic Programming)
Permissive nominal terms resemble nominal terms, but using infinite and co-infinite sets of atoms. We get ‘always fresh’ and ‘always alpha-rename’ properties familiar from first-and higher- order syntax but absent from nominal terms. See also the journal version
- Author PDF for Permissive Nominal Terms and their Unification
- Proceedings PDF for Permissive Nominal Terms and their Unification
- BibTeX for Permissive Nominal Terms and their Unification @inproceedings{gabbay:perntu, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification}}, booktitle = {Atti del Convegno CILC 09: 24-esimo Convegno Italiano di Logica Computazionale}, editor = {Marco Gavanelli and Fabrizio Riguzzi}, pages = {70--84}, publisher = {GULP (Italian Association for Logic Programming)}, address = {Ferrara, Italy}, year = {2009}, month = jun, url = {https://www.programmazionelogica.it/wp-content/uploads/2014/10/cilc09-proceedings.pdf} }
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Publication type: Journal article
A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
Theoretical Computer Science vol. 410, no. 12-13 pp. 1159–1189 Elsevier
Nominal sets contain names and alpha-equivalence, so what about defining a capture-avoiding substitution action for them? We give a notion of “replace a by y in x” where x and y are any two sets.
This subsumes the AISB08 paper, and is followed by Freshness and name-restriction in sets of traces with names.
- Author PDF for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- Publisher / DOI for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- BibTeX for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets @article{gabbay:stusun, author = {Murdoch J. Gabbay}, title = {{A study of substitution, using nominal techniques and Fraenkel-Mostowski sets}}, journal = {Theoretical Computer Science}, volume = {410}, number = {12-13}, pages = {1159--1189}, publisher = {Elsevier}, year = {2009}, month = mar, doi = {10.1016/j.tcs.2008.11.013}, url = {https://doi.org/10.1016/j.tcs.2008.11.013} }
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Publication type: Technical report
Permissive Nominal Terms and their Unification
School of Mathematical and Computer Sciences, Heriot-Watt University Technical Report no. HW-MACS-TR-0062 49 pp. total Edinburgh, UK
We consider permissive nominal terms, and their unification. Permissive nominal terms closely resemble nominal terms, but they recover these useful ‘always fresh’ and ‘always alpha-rename’ properties, familiar from first-and higher- order syntax. In the permissive world, freshness contexts are elided, and the notion of unifier is based on substitution alone, rather than on nominal terms’ notion of substitution plus freshness conditions.
We prove that expressivity is not lost moving to the permissive case and translate from nominal terms unification into permissive nominal terms unification.
- Author PDF for Permissive Nominal Terms and their Unification
- BibTeX for Permissive Nominal Terms and their Unification @techreport{gabbay:perntu-tr, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive Nominal Terms and their Unification}}, institution = {School of Mathematical and Computer Sciences, Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0062}, address = {Edinburgh, UK}, year = {2009}, month = feb, note = {49 pages}, url = {https://www.gabbay.org.uk/papers/perntu-tr.pdf} }
- Later version DOI for Permissive Nominal Terms and their Unification
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Publication type: Technical report
Permissive nominal terms
INRIA Research Report no. RR-6682 15 pp. total Orsay, France
We consider permissive nominal terms, and their unification. Permissive nominal terms closely resemble nominal terms, but they recover these useful ‘always fresh’ and ‘always alpha-rename’ properties, familiar from first-and higher- order syntax. In the permissive world, freshness contexts are elided, and the notion of unifier is based on substitution alone, rather than on nominal terms’ notion of substitution plus freshness conditions.
- Author PDF for Permissive nominal terms
- HAL record for Permissive nominal terms
- BibTeX for Permissive nominal terms @techreport{gabbay:pernt-tr, author = {Gilles Dowek and Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Permissive nominal terms}}, institution = {INRIA}, series = {Research Report}, number = {RR-6682}, address = {Orsay, France}, year = {2008}, month = oct, note = {15 pages}, url = {https://inria.hal.science/inria-00335115} }
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Publication type: Journal article
Capture-Avoiding Substitution as a Nominal Algebra
Formal Aspects of Computing vol. 20, no. 4-5 pp. 451–479 Springer
Nominal algebra makes it easy to write axioms for capture-avoiding substitution—or does it? Consider the choices: do you want b[a↦X]=b or a#Z⊢Z[a↦X]=Z? Do you want Z[a↦a]=Z or not? In this paper we write down various plausible axioms and prove senses in which they are correct.
This is a journal version of our ICTAC06 paper.
- Author PDF for Capture-Avoiding Substitution as a Nominal Algebra
- Publisher / DOI for Capture-Avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-Avoiding Substitution as a Nominal Algebra @article{gabbay:capasn-jv, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-Avoiding Substitution as a Nominal Algebra}}, journal = {Formal Aspects of Computing}, volume = {20}, number = {4-5}, pages = {451--479}, publisher = {Springer}, year = {2008}, month = jun, doi = {10.1007/s00165-007-0056-1}, url = {https://doi.org/10.1007/s00165-007-0056-1} }
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Publication type: Conference paper
Substitution for Fraenkel-Mostowski foundations
Proceedings of the 2008 AISB Symposium on Computing and Philosophy AISB 2008 Convention: Communication, Interaction and Social Intelligence vol. 12 pp. 65–72 University of Aberdeen, Aberdeen, UK: The Society for the Study of Artificial Intelligence and Simulation of Behaviour ISBN 1-902956-71-0
We present a capture-avoiding substitution action defined in Fraenkel-Mostowski set theory on the entire sets universe. Substitution is commonly understood as an action on syntax (for example when building a logic or a lambda calculus) and thus as a prelude to, but not a part of, mathematical denotations. We show that it also has denotational realisation, as an operation on elements of a mathematical foundation.
- Author PDF for Substitution for Fraenkel-Mostowski foundations
- Proceedings PDF for Substitution for Fraenkel-Mostowski foundations
- BibTeX for Substitution for Fraenkel-Mostowski foundations @inproceedings{gabbay:subfmf, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Substitution for Fraenkel-Mostowski foundations}}, booktitle = {Proceedings of the 2008 AISB Symposium on Computing and Philosophy}, series = {AISB 2008 Convention: Communication, Interaction and Social Intelligence}, volume = {12}, pages = {65--72}, publisher = {The Society for the Study of Artificial Intelligence and Simulation of Behaviour}, address = {University of Aberdeen, Aberdeen, UK}, year = {2008}, month = apr, isbn = {1-902956-71-0}, url = {https://aisb.org.uk/wp-content/uploads/2019/12/Final-vol-12.pdf} }
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Publication type: Journal article
Hierarchical Nominal Terms and Their Theory of Rewriting
Electronic Notes in Theoretical Computer Science vol. 174, no. 5 pp. 37–52 Elsevier ISSN 1571-0661
So nominal terms have two levels of variable. But that just means we have internalised the meta-level once. What if we do this repeatedly until we reach a fixedpoint? We build a theory of rewriting on nominal terms with an infinite hierarchy of variables. We examine case studies, including capture- and non-capture avoiding substitution and the NEW calculus of contexts as rewrite systems.
This is a revised and expanded version of my LFMTP06 workshop paper.
- Author PDF for Hierarchical Nominal Terms and Their Theory of Rewriting
- Publisher / DOI for Hierarchical Nominal Terms and Their Theory of Rewriting
- BibTeX for Hierarchical Nominal Terms and Their Theory of Rewriting @article{gabbay:hienr, author = {Murdoch J. Gabbay}, title = {{Hierarchical Nominal Terms and Their Theory of Rewriting}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {174}, number = {5}, pages = {37--52}, publisher = {Elsevier}, year = {2007}, month = jun, doi = {10.1016/j.entcs.2007.01.017}, url = {https://doi.org/10.1016/j.entcs.2007.01.017} }
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Publication type: Journal article
Nominal rewriting (journal version)
Information and Computation vol. 205, no. 6 pp. 917–965
Rewriting on nominal terms. An example: a#X⊢λa.(Xa)→X (eta-contraction). Nominal techniques changed from being about syntax-with-binding to being about specifications-with-binding. We also consider criteria for confluence. See the PPDP04 conference paper and Nominal universal algebra and Closed nominal rewriting and efficiently computable nominal algebra equality.
- Author PDF for Nominal rewriting (journal version)
- Publisher / DOI for Nominal rewriting (journal version)
- BibTeX for Nominal rewriting (journal version) @article{gabbay:nomr-jv, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Nominal rewriting (journal version)}}, journal = {Information and Computation}, volume = {205}, number = {6}, pages = {917--965}, year = {2007}, month = jun, doi = {10.1016/j.ic.2006.12.002}, url = {https://doi.org/10.1016/j.ic.2006.12.002} }
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Publication type: Technical report
Capture-Avoiding Substitution as a Nominal Algebra
Heriot-Watt University HW-MACS-TR-0053
Subsumed by “Capture-Avoiding Substitution as a Nominal Algebra”.
Decidability and theory of models of a nominal algebra axiomatisation of capture-avoiding substitution.
- Author PDF for Capture-Avoiding Substitution as a Nominal Algebra
- Institutional record for Capture-Avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-Avoiding Substitution as a Nominal Algebra @techreport{gabbay:capasn-tr, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-Avoiding Substitution as a Nominal Algebra}}, institution = {Heriot-Watt University}, number = {HW-MACS-TR-0053}, year = {2007}, url = {https://research.tue.nl/en/publications/capture-avoiding-substitution-as-a-nominal-algebra-3/} }
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Publication type: Conference paper
Capture-avoiding Substitution as a Nominal Algebra
Proceedings of the 3rd International Colloquium on Theoretical Aspects of Computing (ICTAC 2006) Lecture Notes in Computer Science vol. 4281 pp. 198–212 Berlin: Springer
We give a detailed investigation of the fundamental operation which is capture-avoiding substitution, investigated from the point of view of Nominal Algebra. See also the journal version.
- Author PDF for Capture-avoiding Substitution as a Nominal Algebra
- Publisher / DOI for Capture-avoiding Substitution as a Nominal Algebra
- BibTeX for Capture-avoiding Substitution as a Nominal Algebra @inproceedings{gabbay:capasn, author = {Murdoch J. Gabbay and Aad Mathijssen}, title = {{Capture-avoiding Substitution as a Nominal Algebra}}, booktitle = {Proceedings of the 3rd International Colloquium on Theoretical Aspects of Computing (ICTAC 2006)}, series = {Lecture Notes in Computer Science}, volume = {4281}, pages = {198--212}, publisher = {Springer}, year = {2006}, month = nov, doi = {10.1007/11921240_14}, url = {https://doi.org/10.1007/11921240_14} }
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Publication type: Conference paper
Hierarchical nominal terms and their theory of rewriting
Proceedings of LFMTP’06: First International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice eds. Alberto Momigliano, Brigitte Pientka pp. 32–45 Seattle, Washington, USA
- Proceedings PDF for Hierarchical nominal terms and their theory of rewriting
- BibTeX for Hierarchical nominal terms and their theory of rewriting @inproceedings{gabbay:hienr-preproceedings, author = {Murdoch J. Gabbay}, title = {{Hierarchical nominal terms and their theory of rewriting}}, booktitle = {Proceedings of LFMTP'06: First International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice}, editor = {Alberto Momigliano and Brigitte Pientka}, pages = {32--45}, address = {Seattle, Washington, USA}, year = {2006}, month = aug, url = {https://www.cs.mcgill.ca/~bpientka/lfmtp.pdf} }
- Later version DOI for Hierarchical nominal terms and their theory of rewriting
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Publication type: Conference paper
Nominal rewriting with name generation: abstraction vs. locality
Proceedings of the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2005) pp. 47–58 ACM Press
We generalise and extend previous work on rewriting with variable binding, with name generation/restriction, and study its properties.
- Author PDF for Nominal rewriting with name generation: abstraction vs. locality
- Publisher / DOI for Nominal rewriting with name generation: abstraction vs. locality
- BibTeX for Nominal rewriting with name generation: abstraction vs. locality @inproceedings{gabbay:nomrng, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay}, title = {{Nominal rewriting with name generation: abstraction vs. locality}}, booktitle = {Proceedings of the 7th ACM SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP 2005)}, pages = {47--58}, publisher = {ACM Press}, year = {2005}, month = jul, doi = {10.1145/1069774.1069779}, url = {https://doi.org/10.1145/1069774.1069779} }
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Publication type: Journal article
Nominal Unification
Theoretical Computer Science vol. 323, no. 1–3 pp. 473–497 Essex, UK: Elsevier ISSN 0304-3975
This paper introduced nominal terms, which extend first-order terms with term-formers that can bind. Unification of nominal terms is decidable and has most general unifiers. This journal paper subsumes a conference version. The next step up in abstraction from nominal terms is their permissive variant.
- Author PDF for Nominal Unification
- Publisher / DOI for Nominal Unification
- BibTeX for Nominal Unification @article{gabbay:nomu-jv, author = {Christian Urban and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{Nominal Unification}}, journal = {Theoretical Computer Science}, volume = {323}, number = {1--3}, pages = {473--497}, publisher = {Elsevier}, year = {2004}, month = sep, doi = {10.1016/j.tcs.2004.06.016}, url = {https://doi.org/10.1016/j.tcs.2004.06.016} }
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Publication type: Conference paper
Nominal Rewriting Systems
Proceedings of the 6th ACM SIGPLAN symposium on Principles and Practice of Declarative Programming (PPDP 2004) pp. 108–119 ACM Press
We generalise first-order rewriting with terms involving binding operations, using a nominal approach in which bound entities are explicitly named.
- Author PDF for Nominal Rewriting Systems
- Publisher / DOI for Nominal Rewriting Systems
- BibTeX for Nominal Rewriting Systems @inproceedings{gabbay:nomr, author = {Maribel Fern{\'a}ndez and Murdoch J. Gabbay and Ian Mackie}, title = {{Nominal Rewriting Systems}}, booktitle = {Proceedings of the 6th ACM SIGPLAN symposium on Principles and Practice of Declarative Programming (PPDP 2004)}, pages = {108--119}, publisher = {ACM Press}, year = {2004}, month = aug, doi = {10.1145/1013963.1013978}, url = {https://doi.org/10.1145/1013963.1013978} }
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Publication type: Conference paper
Nominal Unification
Proceedings of the 17th International Workshop on Computer Science Logic (CSL 2003) Lecture Notes in Computer Science vol. 2803 pp. 513–527 Springer
See here for formal verification of the results Urban in Isabelle.
- Author PDF for Nominal Unification
- Publisher / DOI for Nominal Unification
- BibTeX for Nominal Unification @inproceedings{gabbay:nomu, author = {Christian Urban and Andrew M. Pitts and Murdoch J. Gabbay}, title = {{Nominal Unification}}, booktitle = {Proceedings of the 17th International Workshop on Computer Science Logic (CSL 2003)}, series = {Lecture Notes in Computer Science}, volume = {2803}, pages = {513--527}, publisher = {Springer}, year = {2003}, month = dec, doi = {10.1007/978-3-540-45220-1_41}, url = {https://doi.org/10.1007/978-3-540-45220-1_41} }
Semantics and duality 24 publications
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Publication type: Book chapter
Decentralized Collaborative Action: Cryptoeconomics in Space
Cryptoeconomic Theory eds. Melanie Swan, Soichiro Takagi, Frank Witte Advances in Science and Technology in the Age of AI vol. 1 pp. 53–70 Singapore: World Scientific ISBN 978-1-80061-865-7 eISBN 978-1-80061-867-1
- Author PDF for Decentralized Collaborative Action: Cryptoeconomics in Space
- Publisher / DOI for Decentralized Collaborative Action: Cryptoeconomics in Space
- BibTeX for Decentralized Collaborative Action: Cryptoeconomics in Space @incollection{gabbay:cryptoeconomics-in-space, author = {Murdoch J. Gabbay}, title = {{Decentralized Collaborative Action: Cryptoeconomics in Space}}, booktitle = {Cryptoeconomic Theory}, editor = {Melanie Swan and Soichiro Takagi and Frank Witte}, series = {Advances in Science and Technology in the Age of AI}, volume = {1}, pages = {53--70}, publisher = {World Scientific}, address = {Singapore}, year = {2026}, month = may, isbn = {978-1-80061-865-7}, note = {Electronic ISBN 978-1-80061-867-1}, doi = {10.1142/9781800618664_0004}, eprint = {2504.12493}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.1142/9781800618664_0004} }
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Publication type: Journal article
Semiframes: the algebra of semitopologies and actionable coalitions
Mathematical Structures in Computer Science vol. 36 article e3 Cambridge University Press
- Author PDF for Semiframes: the algebra of semitopologies and actionable coalitions
- Publisher / DOI for Semiframes: the algebra of semitopologies and actionable coalitions
- BibTeX for Semiframes: the algebra of semitopologies and actionable coalitions @article{gabbay:semiframes, author = {Murdoch J. Gabbay}, title = {{Semiframes: the algebra of semitopologies and actionable coalitions}}, journal = {Mathematical Structures in Computer Science}, volume = {36}, eid = {e3}, publisher = {Cambridge University Press}, year = {2026}, month = feb, note = {Article e3}, doi = {10.1017/S096012952510042X}, eprint = {2310.00956}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1017/S096012952510042X} }
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Journal article
Arithmetising Logic: Polynomial Semantics of FOL
Journal of Applied Logics vol. 12, no. 6 pp. 1481–1548 London: College Publications ISSN 2631-9810
- Author PDF for Arithmetising Logic: Polynomial Semantics of FOL
- Journal issue PDF for Arithmetising Logic: Polynomial Semantics of FOL
- BibTeX for Arithmetising Logic: Polynomial Semantics of FOL @article{gabbay:arithmetising-logic, author = {Murdoch J. Gabbay}, title = {{Arithmetising Logic: Polynomial Semantics of FOL}}, journal = {Journal of Applied Logics}, volume = {12}, number = {6}, pages = {1481--1548}, publisher = {College Publications}, address = {London}, year = {2025}, month = oct, issn = {2631-9810}, eprint = {2024/954}, url = {https://www.collegepublications.co.uk/downloads/ifcolog00074.pdf} }
- Repository record for Arithmetising Logic: Polynomial Semantics of FOL
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Publication type: Journal article
Semitopology: a topological approach to decentralized collaborative action
Journal of Logic and Computation vol. 35, no. 5 article exae050 Oxford University Press online
- Author PDF for Semitopology: a topological approach to decentralized collaborative action
- Publisher / DOI for Semitopology: a topological approach to decentralized collaborative action
- BibTeX for Semitopology: a topological approach to decentralized collaborative action @article{gabbay:semitopology-jlc, author = {Murdoch J. Gabbay}, title = {{Semitopology: a topological approach to decentralized collaborative action}}, journal = {Journal of Logic and Computation}, volume = {35}, number = {5}, eid = {exae050}, publisher = {Oxford University Press}, year = {2025}, month = jul, note = {Article exae050}, doi = {10.1093/logcom/exae050}, eprint = {2303.09287}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1093/logcom/exae050} }
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Publication type: Book
Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
Studies in Logic vol. 108 320 pp. total London: College Publications edition 1 ISBN 978-1-84890-465-1
- Author PDF for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- Publisher page for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- BibTeX for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic @book{gabbay:semitopology-monograph, author = {Murdoch J. Gabbay}, title = {{Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic}}, volume = {108}, publisher = {College Publications}, address = {London}, edition = {1}, year = {2024}, month = aug, isbn = {978-1-84890-465-1}, note = {320 pages}, eprint = {2402.03253}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://www.collegepublications.co.uk/logic/?00056=} }
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Publication type: Journal article
Algebras of UTxO blockchains
Mathematical Structures in Computer Science vol. 31, no. 9 In Homage to Martin Hofmann pp. 1034–1089 Cambridge University Press issue dated online
We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.
- Author PDF for Algebras of UTxO blockchains
- Publisher / DOI for Algebras of UTxO blockchains
- BibTeX for Algebras of UTxO blockchains @article{gabbay:algub, author = {Murdoch J. Gabbay}, title = {{Algebras of UTxO blockchains}}, journal = {Mathematical Structures in Computer Science}, volume = {31}, number = {9}, pages = {1034--1089}, publisher = {Cambridge University Press}, year = {2022}, month = jan, note = {Special issue: In Homage to Martin Hofmann}, doi = {10.1017/S0960129521000438}, url = {https://doi.org/10.1017/S0960129521000438} }
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Publication type: Journal article
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
Annals of Pure and Applied Logic vol. 168, no. 3 pp. 501–621 Elsevier
Everything you wanted to know about semantics of the lambda-calculus using nominal techniques.
- Author PDF for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- Publisher / DOI for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
- BibTeX for Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness @article{gabbay:repdul, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness}}, journal = {Annals of Pure and Applied Logic}, volume = {168}, number = {3}, pages = {501--621}, publisher = {Elsevier}, year = {2017}, month = mar, doi = {10.1016/j.apal.2016.10.001}, url = {https://doi.org/10.1016/j.apal.2016.10.001} }
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Publication type: Journal article
Semantics out of context: nominal absolute denotations for first-order logic and computation
Journal of the ACM vol. 63, no. 3 pp. 1–66 article 25 ISSN 1535-9921
Everything you wanted to know about semantics of first-order logic in nominal sets, topologies, and lattices. A sister paper is Stone duality for first-order logic. Available also on ArXiv.
- Author PDF for Semantics out of context: nominal absolute denotations for first-order logic and computation
- Publisher / DOI for Semantics out of context: nominal absolute denotations for first-order logic and computation
- BibTeX for Semantics out of context: nominal absolute denotations for first-order logic and computation @article{gabbay:semooc, author = {Murdoch J. Gabbay}, title = {{Semantics out of context: nominal absolute denotations for first-order logic and computation}}, journal = {Journal of the ACM}, volume = {63}, number = {3}, eid = {25}, pages = {1--66}, year = {2016}, month = jun, doi = {10.1145/2700819}, url = {https://doi.org/10.1145/2700819} }
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Publication type: Journal article
A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
The IfCoLog Journal of Logics and their Applications vol. 1, no. 2 pp. 83–106 College Publications ISBN 978-1-84890-163-6 ISSN 2055-3706
- Repository PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- Journal issue PDF for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- DOI 10.17863/CAM.42150 for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality
- BibTeX for A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality @article{gabbay:simcmt, author = {Michael J. Gabbay and Murdoch J. Gabbay}, title = {{A Simple and Complete Model Theory for Intensional and Extensional Untyped Lambda-Equality}}, journal = {The IfCoLog Journal of Logics and their Applications}, volume = {1}, number = {2}, pages = {83--106}, publisher = {College Publications}, year = {2014}, month = nov, isbn = {978-1-84890-163-6}, issn = {2055-3706}, doi = {10.17863/CAM.42150}, url = {https://doi.org/10.17863/CAM.42150} }
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Publication type: Book chapter
Stone duality for First-Order Logic: a nominal approach
HOWARD-60. A Festschrift on the Occasion of Howard Barringer's 60th Birthday eds. Andrei Voronkov, Margarita Korovina EPiC Series in Computing vol. 42 pp. 178–209 EasyChair
Stone duality for first-order logic. Open predicates map to open sets. No valuations. See also the ancestor of this paper, Stone duality for nominal Boolean algebras with NEW. This version of the paper differs from the version that appeared in the Festschrift in the correction of some small typos and errors.
- Author PDF for Stone duality for First-Order Logic: a nominal approach
- Publisher / DOI for Stone duality for First-Order Logic: a nominal approach
- BibTeX for Stone duality for First-Order Logic: a nominal approach @incollection{gabbay:stodfo, author = {Murdoch J. Gabbay}, title = {{Stone duality for First-Order Logic: a nominal approach}}, booktitle = {HOWARD-60. A Festschrift on the Occasion of Howard Barringer's 60th Birthday}, editor = {Andrei Voronkov and Margarita Korovina}, series = {EPiC Series in Computing}, volume = {42}, pages = {178--209}, publisher = {EasyChair}, year = {2014}, month = feb, doi = {10.29007/tp3z}, url = {https://doi.org/10.29007/tp3z} }
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Publication type: Journal article
Imaginary groups: lazy monoids and reversible computation.
Mathematical Structures in Computer Science vol. 23, no. 05 pp. 1002–1031 Cambridge University Press
A duality between partially ordered groups and an (apparently novel) category of partially ordered monoids and lazy homomorphisms between them. Every monoid can be lazily injected into a group. Reading a monoid as an abstract notion of computation (or at least, of composition), this construction shows how to add reversibility.
- Author PDF for Imaginary groups: lazy monoids and reversible computation.
- Publisher / DOI for Imaginary groups: lazy monoids and reversible computation.
- BibTeX for Imaginary groups: lazy monoids and reversible computation. @article{gabbay:imaglm, author = {Murdoch J. Gabbay and Peter Kropholler}, title = {{Imaginary groups: lazy monoids and reversible computation.}}, journal = {Mathematical Structures in Computer Science}, volume = {23}, number = {05}, pages = {1002--1031}, publisher = {Cambridge University Press}, year = {2013}, month = oct, doi = {10.1017/S0960129512000849}, url = {https://doi.org/10.1017/S0960129512000849} }
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Publication type: Journal article
Denotation of contextual modal type theory (CMTT): syntax and meta-programming
Journal of Applied Logic vol. 11, no. 1 pp. 1–29
A denotation for CMTT. I wanted to understand this for years, and now we have an answer!
- Author PDF for Denotation of contextual modal type theory (CMTT): syntax and meta-programming
- Publisher / DOI for Denotation of contextual modal type theory (CMTT): syntax and meta-programming
- BibTeX for Denotation of contextual modal type theory (CMTT): syntax and meta-programming @article{gabbay:dencmt, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of contextual modal type theory (CMTT): syntax and meta-programming}}, journal = {Journal of Applied Logic}, volume = {11}, number = {1}, pages = {1--29}, year = {2013}, month = mar, doi = {10.1016/j.jal.2012.07.002}, url = {https://doi.org/10.1016/j.jal.2012.07.002} }
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Publication type: Journal article
Game semantics in the nominal model
Electronic Notes in Theoretical Computer Science vol. 286 Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012) pp. 173–189 Elsevier
Nominal semantics used to model and generalise pointer sequences, using a nice new idea of nominal coabstraction. The result: an expression of game semantics using nominal-style atoms.
- Author PDF for Game semantics in the nominal model
- Publisher / DOI for Game semantics in the nominal model
- BibTeX for Game semantics in the nominal model @article{gabbay:gamsnm, author = {Murdoch J. Gabbay and Dan R. Ghica}, title = {{Game semantics in the nominal model}}, journal = {Electronic Notes in Theoretical Computer Science}, volume = {286}, pages = {173--189}, publisher = {Elsevier}, year = {2012}, month = sep, note = {Special issue: Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2012)}, doi = {10.1016/j.entcs.2012.08.012}, url = {https://doi.org/10.1016/j.entcs.2012.08.012} }
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Publication type: Conference paper
Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
Proceedings of the 9th Italian Convention on Computational Logic (CILC 2012) ed. Francesca A. Lisi CEUR workshop proceedings vol. 857 pp. 104–118 CEUR-WS.org
Universal lattice properties used to specify top, meet, and forall as instances of a single notion of fresh-finite limit.
- Author PDF for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
- Proceedings record for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits
- BibTeX for Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits @inproceedings{gabbay:nomspl, author = {Gilles Dowek and Murdoch J. Gabbay}, title = {{Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits}}, booktitle = {Proceedings of the 9th Italian Convention on Computational Logic (CILC 2012)}, editor = {Francesca A. Lisi}, series = {CEUR workshop proceedings}, volume = {857}, pages = {104--118}, publisher = {CEUR-WS.org}, year = {2012}, month = may, url = {https://nbn-resolving.de/urn:nbn:de:0074-857-8} }
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Publication type: Preprint
Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
Subsumed by “Denotation of contextual modal type theory (CMTT): syntax and meta-programming”.
- Author PDF for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- arXiv record (1202.0904) for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)
- BibTeX for Denotation of syntax and metaprogramming in contextual modal type theory (CMTT) @misc{gabbay:densmc, author = {Murdoch J. Gabbay and Aleksandar Nanevski}, title = {{Denotation of syntax and metaprogramming in contextual modal type theory (CMTT)}}, howpublished = {arXiv preprint arXiv:1202.0904}, year = {2012}, month = feb, eprint = {1202.0904}, archivePrefix = {arXiv}, url = {https://arxiv.org/abs/1202.0904} }
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Publication type: Journal article
Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
Mathematical Structures in Computer Science vol. 21, no. 5 pp. 997–1033 Cambridge University Press
A nominal denotational semantics for meta-variables. Nominal techniques have already given semantics to variables; in this paper we show how they can do the same for meta-variables. The version on this webpage includes corrections.
- Author PDF for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- Publisher / DOI for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables
- BibTeX for Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables @article{gabbay:twolns, author = {Murdoch J. Gabbay}, title = {{Two-level nominal sets and semantic nominal terms: an extension of nominal set theory for handling meta-variables}}, journal = {Mathematical Structures in Computer Science}, volume = {21}, number = {5}, pages = {997--1033}, publisher = {Cambridge University Press}, year = {2011}, month = oct, doi = {10.1017/S0960129511000272}, url = {https://doi.org/10.1017/S0960129511000272} }
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Publication type: Conference paper
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011) EPTCS vol. 71 pp. 58–75
Henkin-style models of the lambda-calculus in nominal sets. No Tarski-style valuations; instead we use nominal substitution algebras. The title of the paper in the pre-proceedings was Semantics for functions of the simply-typed lambda-calculus with nominal sets.
- Author PDF for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- Publisher / DOI for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
- BibTeX for Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets @inproceedings{gabbay:nomhss, author = {Murdoch J. Gabbay and Dominic P. Mulligan}, title = {{Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets}}, booktitle = {Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP 2011)}, series = {EPTCS}, volume = {71}, pages = {58--75}, year = {2011}, month = sep, doi = {10.4204/EPTCS.71.5}, url = {https://doi.org/10.4204/EPTCS.71.5} }
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Publication type: Conference paper
Stone duality for nominal Boolean algebras with NEW
Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011) Lecture Notes in Computer Science vol. 6859 pp. 192–207 Springer ISBN 978-3-642-22943-5
The NEW quantifier is the canonical name-generation/name-restriction operator for predicates on the nominal model of names and binding. We give a sets representation theorem and Stone-type duality. See also the subsequent Stone duality for first-order logic.
- Author PDF for Stone duality for nominal Boolean algebras with NEW
- Publisher / DOI for Stone duality for nominal Boolean algebras with NEW
- BibTeX for Stone duality for nominal Boolean algebras with NEW @inproceedings{gabbay:stodnb, author = {Murdoch J. Gabbay and Tadeusz Litak and Daniela Petri{\c{s}}an}, title = {{Stone duality for nominal Boolean algebras with NEW}}, booktitle = {Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011)}, series = {Lecture Notes in Computer Science}, volume = {6859}, pages = {192--207}, publisher = {Springer}, year = {2011}, month = aug, doi = {10.1007/978-3-642-22944-2_14}, url = {https://doi.org/10.1007/978-3-642-22944-2_14} }
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Publication type: Journal article
Foundations of nominal techniques: logic and semantics of variables in abstract syntax
Bulletin of Symbolic Logic vol. 17, no. 2 pp. 161–229
A survey and update paper describing nominal techniques, with specific application to nominal abstract syntax. A sequel to this paper is Nominal terms and nominal logics.
- Author PDF for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- Publisher / DOI for Foundations of nominal techniques: logic and semantics of variables in abstract syntax
- BibTeX for Foundations of nominal techniques: logic and semantics of variables in abstract syntax @article{gabbay:fountl, author = {Murdoch J. Gabbay}, title = {{Foundations of nominal techniques: logic and semantics of variables in abstract syntax}}, journal = {Bulletin of Symbolic Logic}, volume = {17}, number = {2}, pages = {161--229}, year = {2011}, month = jun, doi = {10.2178/bsl/1305810911}, url = {https://doi.org/10.2178/bsl/1305810911} }
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Publication type: Journal article
A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
Theoretical Computer Science vol. 410, no. 12-13 pp. 1159–1189 Elsevier
Nominal sets contain names and alpha-equivalence, so what about defining a capture-avoiding substitution action for them? We give a notion of “replace a by y in x” where x and y are any two sets.
This subsumes the AISB08 paper, and is followed by Freshness and name-restriction in sets of traces with names.
- Author PDF for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- Publisher / DOI for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets
- BibTeX for A study of substitution, using nominal techniques and Fraenkel-Mostowski sets @article{gabbay:stusun, author = {Murdoch J. Gabbay}, title = {{A study of substitution, using nominal techniques and Fraenkel-Mostowski sets}}, journal = {Theoretical Computer Science}, volume = {410}, number = {12-13}, pages = {1159--1189}, publisher = {Elsevier}, year = {2009}, month = mar, doi = {10.1016/j.tcs.2008.11.013}, url = {https://doi.org/10.1016/j.tcs.2008.11.013} }
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Publication type: Conference paper
Nominal renaming sets
Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008) pp. 158–173 Springer
Nominal renaming sets are like nominal sets, but they are based on a finitely-supported renaming action — a renaming is a function from atoms to atoms, not necessarily injective. This provides a convenient sets-based semantics with which to justify Higher-Order Abstract Syntax.
- Author PDF for Nominal renaming sets
- Publisher / DOI for Nominal renaming sets
- BibTeX for Nominal renaming sets @inproceedings{gabbay:nomrs, author = {Murdoch J. Gabbay and Martin Hofmann}, title = {{Nominal renaming sets}}, booktitle = {Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2008)}, pages = {158--173}, publisher = {Springer}, year = {2008}, month = nov, doi = {10.1007/978-3-540-89439-1_11}, url = {https://doi.org/10.1007/978-3-540-89439-1_11} }
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Publication type: Conference paper
Substitution for Fraenkel-Mostowski foundations
Proceedings of the 2008 AISB Symposium on Computing and Philosophy AISB 2008 Convention: Communication, Interaction and Social Intelligence vol. 12 pp. 65–72 University of Aberdeen, Aberdeen, UK: The Society for the Study of Artificial Intelligence and Simulation of Behaviour ISBN 1-902956-71-0
We present a capture-avoiding substitution action defined in Fraenkel-Mostowski set theory on the entire sets universe. Substitution is commonly understood as an action on syntax (for example when building a logic or a lambda calculus) and thus as a prelude to, but not a part of, mathematical denotations. We show that it also has denotational realisation, as an operation on elements of a mathematical foundation.
- Author PDF for Substitution for Fraenkel-Mostowski foundations
- Proceedings PDF for Substitution for Fraenkel-Mostowski foundations
- BibTeX for Substitution for Fraenkel-Mostowski foundations @inproceedings{gabbay:subfmf, author = {Murdoch J. Gabbay and Michael J. Gabbay}, title = {{Substitution for Fraenkel-Mostowski foundations}}, booktitle = {Proceedings of the 2008 AISB Symposium on Computing and Philosophy}, series = {AISB 2008 Convention: Communication, Interaction and Social Intelligence}, volume = {12}, pages = {65--72}, publisher = {The Society for the Study of Artificial Intelligence and Simulation of Behaviour}, address = {University of Aberdeen, Aberdeen, UK}, year = {2008}, month = apr, isbn = {1-902956-71-0}, url = {https://aisb.org.uk/wp-content/uploads/2019/12/Final-vol-12.pdf} }
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Publication type: Technical report
Nominal renaming sets
Heriot-Watt University Technical Report no. HW-MACS-TR-0058 35 pp. total Edinburgh, UK
Subsumed by my LPAR08 paper with Martin Hofmann. Nominal sets based on a renaming action (possibly non-injective functions on atoms) instead of the more usual permutation action.
- Author PDF for Nominal renaming sets
- BibTeX for Nominal renaming sets @techreport{gabbay:nomrs-tr, author = {Murdoch J. Gabbay}, title = {{Nominal renaming sets}}, institution = {Heriot-Watt University}, series = {Technical Report}, number = {HW-MACS-TR-0058}, address = {Edinburgh, UK}, year = {2007}, month = dec, note = {35 pages}, url = {https://www.gabbay.org.uk/papers/nomrs-tr.pdf} }
- Later version DOI for Nominal renaming sets
Semitopology and semiframes 5 publications
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Publication type: Book chapter
Decentralized Collaborative Action: Cryptoeconomics in Space
Cryptoeconomic Theory eds. Melanie Swan, Soichiro Takagi, Frank Witte Advances in Science and Technology in the Age of AI vol. 1 pp. 53–70 Singapore: World Scientific ISBN 978-1-80061-865-7 eISBN 978-1-80061-867-1
- Author PDF for Decentralized Collaborative Action: Cryptoeconomics in Space
- Publisher / DOI for Decentralized Collaborative Action: Cryptoeconomics in Space
- BibTeX for Decentralized Collaborative Action: Cryptoeconomics in Space @incollection{gabbay:cryptoeconomics-in-space, author = {Murdoch J. Gabbay}, title = {{Decentralized Collaborative Action: Cryptoeconomics in Space}}, booktitle = {Cryptoeconomic Theory}, editor = {Melanie Swan and Soichiro Takagi and Frank Witte}, series = {Advances in Science and Technology in the Age of AI}, volume = {1}, pages = {53--70}, publisher = {World Scientific}, address = {Singapore}, year = {2026}, month = may, isbn = {978-1-80061-865-7}, note = {Electronic ISBN 978-1-80061-867-1}, doi = {10.1142/9781800618664_0004}, eprint = {2504.12493}, archivePrefix = {arXiv}, primaryClass = {cs.CR}, url = {https://doi.org/10.1142/9781800618664_0004} }
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Publication type: Journal article
Semiframes: the algebra of semitopologies and actionable coalitions
Mathematical Structures in Computer Science vol. 36 article e3 Cambridge University Press
- Author PDF for Semiframes: the algebra of semitopologies and actionable coalitions
- Publisher / DOI for Semiframes: the algebra of semitopologies and actionable coalitions
- BibTeX for Semiframes: the algebra of semitopologies and actionable coalitions @article{gabbay:semiframes, author = {Murdoch J. Gabbay}, title = {{Semiframes: the algebra of semitopologies and actionable coalitions}}, journal = {Mathematical Structures in Computer Science}, volume = {36}, eid = {e3}, publisher = {Cambridge University Press}, year = {2026}, month = feb, note = {Article e3}, doi = {10.1017/S096012952510042X}, eprint = {2310.00956}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1017/S096012952510042X} }
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Publication type: Preprint
Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
arXiv latest revision
- Author PDF for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- arXiv for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies
- BibTeX for Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies @misc{gabbay:declarative-distributed-semitopologies, author = {Murdoch J. Gabbay}, title = {{Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies}}, howpublished = {arXiv preprint arXiv:2512.21137}, year = {2025}, month = dec, doi = {10.48550/arXiv.2512.21137}, eprint = {2512.21137}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.48550/arXiv.2512.21137} }
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Publication type: Journal article
Semitopology: a topological approach to decentralized collaborative action
Journal of Logic and Computation vol. 35, no. 5 article exae050 Oxford University Press online
- Author PDF for Semitopology: a topological approach to decentralized collaborative action
- Publisher / DOI for Semitopology: a topological approach to decentralized collaborative action
- BibTeX for Semitopology: a topological approach to decentralized collaborative action @article{gabbay:semitopology-jlc, author = {Murdoch J. Gabbay}, title = {{Semitopology: a topological approach to decentralized collaborative action}}, journal = {Journal of Logic and Computation}, volume = {35}, number = {5}, eid = {exae050}, publisher = {Oxford University Press}, year = {2025}, month = jul, note = {Article exae050}, doi = {10.1093/logcom/exae050}, eprint = {2303.09287}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://doi.org/10.1093/logcom/exae050} }
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Publication type: Book
Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
Studies in Logic vol. 108 320 pp. total London: College Publications edition 1 ISBN 978-1-84890-465-1
- Author PDF for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- Publisher page for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic
- BibTeX for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic @book{gabbay:semitopology-monograph, author = {Murdoch J. Gabbay}, title = {{Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic}}, volume = {108}, publisher = {College Publications}, address = {London}, edition = {1}, year = {2024}, month = aug, isbn = {978-1-84890-465-1}, note = {320 pages}, eprint = {2402.03253}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://www.collegepublications.co.uk/logic/?00056=} }
Publications by title
Publications by type
Book
Semitopology
Decentralised Collaborative Action via Topology, Algebra, and Logic
Semitopologies provide a mathematical foundation for heterogeneous, decentralised, permissionless computing systems. Points stand for participants; open sets describe ways in which they can cooperate.
- Published
- August 2024
- ISBN
- 978-1-84890-465-1
- Print · UK
- Print · US
- Digital edition
- Book preprint · PDF
- BibTeX for Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic @book{gabbay:semitopology-monograph, author = {Murdoch J. Gabbay}, title = {{Semitopology: Decentralised Collaborative Action via Topology, Algebra, and Logic}}, volume = {108}, publisher = {College Publications}, address = {London}, edition = {1}, year = {2024}, month = aug, isbn = {978-1-84890-465-1}, note = {320 pages}, eprint = {2402.03253}, archivePrefix = {arXiv}, primaryClass = {cs.LO}, url = {https://www.collegepublications.co.uk/logic/?00056=} }
Talks
Thanks to the seminar organisers who gave me opportunities to present the ideas below, and thanks to all of those who made time to turn up.
Selected talks
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Talk type: Conference talk
A declarative approach to specifying distributed algorithms using three-valued modal logic
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Conference talk
A shallow arithmetisation of first-order validity
I discuss how to directly arithmetise (cryptographer-speak for ‘translate into polynomials, the language of cryptography’) validity over finite models. This means that you can specify a property – e.g. some desired correctness property – and via cryptographic magic generate succinct or zero-knowledge proofs that a model satisfies that correctness property, without communicating that model to the recipient of the proof. This is very useful, I would argue, for quickly proving correctness of data, behaviour, or anything else that can be finitely represented.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Workshop talk
Semitopologies and semiframes for distributed heterogeneous consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). Semitopologies naturally generate a three-valued logic (by considering continuous functions out of them), and they can be dualised to semiframes, which are complete join-semilattices with a compatibility relation. We study these, along with various well-behavedness conditions on points in a semitopology.
Presented by Murdoch J. Gabbay First Workshop on Heterogeneous Trust in Distributed Systems (HTDS 2023) Convocation Room 113, Friend Center Princeton, New Jersey, USA
- Slides for Semitopologies and semiframes for distributed heterogeneous consensus
- Recording of Semitopologies and semiframes for distributed heterogeneous consensus
- Related paper for Semitopologies and semiframes for distributed heterogeneous consensus
- Semiframes paper for Semitopologies and semiframes for distributed heterogeneous consensus
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Talk type: Conference talk
Nominal techniques as an Agda library
Nominal techniques … in Agda! Proof-of-concept but a promising start. If you like this and want to develop it further or transfer the ideas to (say) Lean, then please get in touch.
Presented by Orestis Melkonian With Murdoch J. Gabbay 29th International Conference on Types for Proofs and Programs (TYPES 2023) ETSI Informàtica, Universitat Politècnica de València Hosted by Universitat Politècnica de València Valencia, Spain
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Talk type: Tech talk
What is an EUTxO blockchain?
A revised and expanded exploration of a new and simple mathematical model of blockchains
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Course / lecture
Nominal Datatypes in Haskell
We demo my Nominal Datatypes Haskell package
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2020 (SPLV20) Laboratory for Foundations of Computer Science, University of Edinburgh Hosted by University of Edinburgh Online
- Nominal Datatypes in Haskell — Lecture 1 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 1
- Nominal Datatypes in Haskell — Lecture 2 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 2
- Nominal Datatypes in Haskell — Lecture 3 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 3
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Talk type: Conference talk
Nominal techniques: past and future
An observation of nominal techniques is that names are a datatype
Thanks to the ICALP organisers. See the ICALP programme.
ICALP 2019 Patras, Greece
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Talk type: Seminar
Topological nominal semantics
Stone duality in a nominal universe.
Thanks to Dan Marsden and Luke Ong. I outline two papers on algebraic and topological semantics for the lambda-calculus and first-order logic. See the announcement and abstract.
OASIS seminar, Department of Computer Science, University of Oxford Oxford, UK
Full talks list
186 items, arranged by date.
Talks by date
2026
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Talk type: Course / lecture
Distributed protocols: a logical approach
Distributed protocols are famously hard to get right, specify, and reason about. Logic can help! For example, it can help us give a three-day summer school course on a message-free, time-free, failure-free logical abstraction for specifying message-passing timed protocols with failure.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2026 (SPLV26) University of Glasgow Glasgow, Scotland
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Talk type: Research talk
Exponential cake and consistent controllers
This talk treats the consistent controllers problem: suppose you have multiple interacting state machines (controllers) A, B, C, and D. Suppose controller B is bad and might equivocate, meaning that it may report different and incompatible states to peers A, C, and D. How can peers A, C, and D detect this misbehaviour and gracefully recover to a consistent state? Answers that invoke a single controlling peer G that suspends A, C, and D and resets them to a most recent consistent state are not admissible; there is no deus ex machina here, just controllers trying to stay consistent. If this problem seems hypothetical – it’s not. The world already consists of interacting controllers and the problem of consistency is a serious brake on safe progress.
Presented by Murdoch J. Gabbay With Isaac Sheff, Anthony Hart Heliax Research Talk
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Talk type: Workshop talk
Declarative Bracha Broadcast using 3-valued logic
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour. The result is simple and highly effective.
Presented by Murdoch J. Gabbay Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
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Talk type: Workshop talk
Heterogeneous Broadcast
A protocol is heterogeneous when different controllers may have different trust assumptions. A controller is a machine that issues state: a bank, a national register, a blockchain, etc. The world has many controllers, each of which a) have their own trust assumptions and b) must interoperate with the others. Multiple trust assumptions takes us to a distinct design space, so that Heterogeneous Bracha Broadcast is not merely a generalisation of (Homogeneous) Bracha Broadcast: it’s a different beast.
Presented by Murdoch J. Gabbay With Isaac Sheff Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
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Talk type: Seminar
Arithmetising FOL
I give an accessible introduction to my paper on Arithmetisation of computation via polynomial semantics for first-order logic in the Journal of Applied Logics, Volume 12, number 6, pp 1481–1548, October 2025.
Presented by Murdoch J. Gabbay Imperial College Research Seminar
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Talk type: Seminar
Declarative distributed protocols using three-valued modal logic and semitopologies
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour, and semitopologies provide a language for describing quorums. The result is to exhibit protocols as simple and compact axiomatic theories.
Presented by Murdoch J. Gabbay Paris Automata and Concurrency Theory Seminar Paris, France
- Slides for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Bracha Broadcast for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Paxos for Declarative distributed protocols using three-valued modal logic and semitopologies
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Talk type: Workshop talk
Declarative distributed protocols using three-valued modal logic types
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 26071: Behavioural Types for Resilience Schloss Dagstuhl Dagstuhl, Germany
2025
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Talk type: Seminar
SNARKs for pure logic
A presentation of zkFOL (arithmetisation of first-order logic) to Circle’s cryptography group.
Presented by Murdoch J. Gabbay Circle Research Seminar
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Talk type: Teaching talk
Jamie’s musings and dubious advice
I dispense dubious life advice to PhD students participating in the SPLV summer school. I can’t guarantee that the advice is sound, but I can guarantee that every mistake listed is one which I have personally seen made (mostly by myself). If only I had known then what I know now, well, I’d probably have made a whole different collection of mistakes. This talk was dedicated to the memory of Gilles Dowek, who was a colleague, friend, and most pertinently to this talk, an inexhaustible source of wise advice.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2025 (SPLV25) University of Edinburgh Edinburgh, Scotland
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Against Cryptoeconomics Day
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Anoma Day
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Talk type: Conference talk
A declarative approach to specifying distributed algorithms using three-valued modal logic
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Conference talk
A shallow arithmetisation of first-order validity
I discuss how to directly arithmetise (cryptographer-speak for ‘translate into polynomials, the language of cryptography’) validity over finite models. This means that you can specify a property – e.g. some desired correctness property – and via cryptographic magic generate succinct or zero-knowledge proofs that a model satisfies that correctness property, without communicating that model to the recipient of the proof. This is very useful, I would argue, for quickly proving correctness of data, behaviour, or anything else that can be finitely represented.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Workshop talk
Message logic for service commitments
Message logic is a logic for specifying and reasoning about promises that participants in a message-passing distributed system can make to one another. This sounds simple but it turns out to lead to a surprisingly nice logical framework. This logic is somewhat different from the logics I’m used to studying, notably because the notion of logical implication has a measure-theoretic flavour. This comes from the fact that promises made might be broken, even in good faith, e.g. if the entire network fails.
Presented by Murdoch J. Gabbay IC3 Winter Retreat
2024
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Talk type: Lightning talk
Polynomial semantics for FOL
I discuss zkFOL, the arithmetisation of first-order logic validity over finite models.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2024 (SPLV24) University of Strathclyde Glasgow, Scotland
2023
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Talk type: Workshop talk
Semitopologies and semiframes for distributed heterogeneous consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). Semitopologies naturally generate a three-valued logic (by considering continuous functions out of them), and they can be dualised to semiframes, which are complete join-semilattices with a compatibility relation. We study these, along with various well-behavedness conditions on points in a semitopology.
Presented by Murdoch J. Gabbay First Workshop on Heterogeneous Trust in Distributed Systems (HTDS 2023) Convocation Room 113, Friend Center Princeton, New Jersey, USA
- Slides for Semitopologies and semiframes for distributed heterogeneous consensus
- Recording of Semitopologies and semiframes for distributed heterogeneous consensus
- Related paper for Semitopologies and semiframes for distributed heterogeneous consensus
- Semiframes paper for Semitopologies and semiframes for distributed heterogeneous consensus
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Talk type: Workshop talk
Semitopology and (paraconsistent) logic
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). It turns out that semitopologies are most easily classified using a three-valued paraconsistent logic; put another way, when you try to develop a logic over semitopologies, with semantics as continuous functions, a third truth-value very naturally appears.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 23272: Epistemic and Topological Reasoning in Distributed Systems Schloss Dagstuhl Dagstuhl, Germany
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Talk type: Seminar
Equivariance: the name of the true monster in your induction
My PhD was all about difficulties encountered combining inductively defined syntax with variable binding. In this talk I make two arguments: first, that names are best considered a primitive datatype with specific mathematical properties; and second, that the specific property is fundamentally not the fancy stuff that most of my PhD was about, but instead it is the apparently bland assertion of equivariance, namely, the property that a name-permutation universally applied, preserves validity. Once you have equivariance, I argue, the rest of the theory drops out. Conversely, if you do not take equivariance as primitive, your treatment of names may become painful. In short: equivariance is it. For more on these ideas, see the associated paper, the Nominal Haskell package, and the Nominal Agda package.
Presented by Murdoch J. Gabbay LAIV seminar
- Slides for Equivariance: the name of the true monster in your induction
- Related paper for Equivariance: the name of the true monster in your induction
- Nominal Haskell package for Equivariance: the name of the true monster in your induction
- Nominal Agda package for Equivariance: the name of the true monster in your induction
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Talk type: Conference talk
Nominal techniques as an Agda library
Nominal techniques … in Agda! Proof-of-concept but a promising start. If you like this and want to develop it further or transfer the ideas to (say) Lean, then please get in touch.
Presented by Orestis Melkonian With Murdoch J. Gabbay 29th International Conference on Types for Proofs and Programs (TYPES 2023) ETSI Informàtica, Universitat Politècnica de València Hosted by Universitat Politècnica de València Valencia, Spain
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Talk type: Seminar
Semitopology: a new topological model of heterogeneous consensus
A quorum in a distributed system is a set of participants with sufficient power to take collective action, independently from the rest of the system. Quorums are closed under arbitrary union, but not under arbitrary intersection. It turns out that if you treat this as something that is nearly (but not quite) a point-set topology, and then replay topological notions like open set, closed set, dense set, and continuous function, interpreted as quorums, then you obtain constructs with real meaning in the theory of distributed systems (e.g. a dense set corresponds to a blocking set). It turns out further that if you take notions from distributed systems and view them topologically, then further rich structure is revealed.
Presented by Murdoch J. Gabbay Scottish Programming Languages Seminar (SPLS) Coats Building, University of the West of Scotland Paisley, Scotland
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Talk type: Tech talk
Karl Marx and the blockchain
This talk is based on a book chapter on Karl Marx and the blockchain coauthored with Devraj Basu (which at least one reviewer really liked) and an article in the Times Higher Education. Marx was an economic thinker and a revolutionary, and it’s his economic thought that the talk and associated article focus on. The article unpacks a case for the relevance of Marxian economic thought to blockchain technology. Writing this was fun, and it was a nice change from pure mathematics.
Presented by Murdoch J. Gabbay Huawei Tech Talk Edinburgh, UK
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Talk type: Colloquium
A broadly Marxian analysis of blockchain and cryptocurrencies
This talk is based on a book chapter on Karl Marx and the blockchain coauthored with Devraj Basu (which at least one reviewer really liked) and an article in the Times Higher Education. Marx was an economic thinker and a revolutionary, and it’s his economic thought that the talk and associated article focus on. The article unpacks a case for the relevance of Marxian economic thought to blockchain technology. Writing this was fun, and it was a nice change from pure mathematics.
Presented by Murdoch J. Gabbay Middlesex Computer Science Colloquium
2022
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Talk type: Seminar
The semitopology of heterogeneous consensus
A mathematical notion of consensus in a heterogeneous permissionless environment. ‘Permissionless’ means that participants are free to join or leave the network without permission from a central authority; ‘heterogeneous’ means that participants may have different goals and beliefs.
Thanks to the Online Social Choice and Welfare seminar series (permalink). Video online here.
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Talk type: Conference talk
The (semi)topology of distributed consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open. This generalises the notion of quorum from distributed systems. It soon becomes clear that the most interesting semitopologies are those that satisfy an intersection-of-open-sets property which is the precise negation of Hausdorff separation.
Presented by Murdoch J. Gabbay 11th International Conference on Geometric and Topological Methods in Computer Science (GETCO 2022) EPITA School of Engineering and Computer Science Le Kremlin-Bicêtre, France / hybrid
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Talk type: Seminar
An overview of my claimed proof of consistency of Quine's New Foundations (ConNF)
As the title suggests. The claimed proof is here.
Thanks to the LFCS seminar series. Video online here.
LFCS seminar series, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
The Cairo Test Suite Workshop
An online presentation of my Cairo property-based and unit test suite (also on GitHub).
Thanks to Matchbox DAO for the kind invitation to speak.
Matchbox DAO Online
2021
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Talk type: Seminar
Nominal programming for the working programmer
What it takes to implement a nominal package in a programming language, drawing on the Nominal Haskell package.
Presented by Murdoch J. Gabbay Scottish Programming Languages Seminar (SPLS) Hosted by University of Strathclyde Online
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Talk type: Seminar
The Design of the Nominal Datatypes Package in Haskell
Introducing: a Nominal monad
Thanks to the Huawei-Edinburgh Joint Lab. Seminar details and video here.
Huawei–Edinburgh Joint Lab Edinburgh, UK
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Talk type: Workshop talk
Equivariant ZFA and the foundations of nominal techniques
On the foundations of nominal techniques
Thanks to the summer school organisers.
XIII Summer Workshop in Mathematics Hosted by Universidade de Brasília Online
2020
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Talk type: Tech talk
A semi-topological view of real-world consensus
Consensus = Topology
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Workshop talk
A semi-topological view of real-world consensus
Consensus = Topology Thanks to the FRIDA organisers.
7th Workshop on Formal Reasoning in Distributed Algorithms (FRIDA 2020) Online
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Talk type: Tech talk
What is an EUTxO blockchain?
A revised and expanded exploration of a new and simple mathematical model of blockchains
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Course / lecture
Nominal Datatypes in Haskell
We demo my Nominal Datatypes Haskell package
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2020 (SPLV20) Laboratory for Foundations of Computer Science, University of Edinburgh Hosted by University of Edinburgh Online
- Nominal Datatypes in Haskell — Lecture 1 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 1
- Nominal Datatypes in Haskell — Lecture 2 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 2
- Nominal Datatypes in Haskell — Lecture 3 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 3
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Talk type: Seminar
What is an EUTxO blockchain?
An exploration of a new and simple mathematical model of blockchains
Thanks to Ohad Kammar. See the LFCS Seminar series videos online.
LFCS seminar series, University of Edinburgh Edinburgh, UK / online
2019
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Talk type: Course / lecture
Programming with nominal techniques
Outline of a nominal datatypes package.
Thanks to the summer school organisers. See the summer school webpage.
International Summer School on Metaprogramming 2019 Schloss Dagstuhl, Germany
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Talk type: Conference talk
Nominal techniques: past and future
An observation of nominal techniques is that names are a datatype
Thanks to the ICALP organisers. See the ICALP programme.
ICALP 2019 Patras, Greece
2018
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Talk type: Seminar
The language of stratified sets, Quine's NF, rewriting, and higher-order logic
A somewhat meandering tour of some elementary and not-so-elementary results
Thanks to Jessica Ryan. See the announcement and abstract.
FATA seminar, Department of Computer Science, University of Glasgow Glasgow, UK
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Talk type: Seminar
Topological nominal semantics
Stone duality in a nominal universe.
Thanks to Dan Marsden and Luke Ong. I outline two papers on algebraic and topological semantics for the lambda-calculus and first-order logic. See the announcement and abstract.
OASIS seminar, Department of Computer Science, University of Oxford Oxford, UK
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Talk type: Workshop talk
Equivariant ZFA with Choice (EZFAC)
I argue for EZFAC as a foundation for nominal techniques.
Thanks to the organisers for this lovely event. Here are the ARW conference proceedings, my poster, and my slides. A longer paper is here.
Automated Reasoning Workshop 2018 Computer Laboratory, Cambridge, UK
2017
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Talk type: Seminar
Imaginary groups
I visit the maths department.
Thanks to the Maximals seminar series.
Maximals seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Course / lecture
Technical writing
A seminar given to local MSc students, on technical writing.
MSc course, Heriot-Watt University Edinburgh, UK
2016
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Just a lovely, friendly event.
Thanks to the SPLS, and to the Departmental Seminar series of Strathclyde.
Scottish Programming Languages Seminar, University of Strathclyde Glasgow, UK
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Talk type: Conference talk
Consistency of Quine's NF using nominal techniques
My first time in Brussels. It seemed an extremly nice place and was surprisingly laid-back. The Highlights Conference was lovely and I will go next year (in London) if I can. In spite of the similar title, this talk was not a pure repeat of my talks at the Logic Colloquium in August (see below). Thanks to the Highlights Conference 2016, and to Slawomir Lasota for inviting me to speak in his session.
Highlights Conference 2016 Brussels, Belgium
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Talk type: Conference talk
Consistency of Quine's NF using nominal techniques
This was the second of two connected presentations; see next talk. Thanks to the Logic Colloquium, and to Nicola Gambino for being a gentleman.
Logic Colloquium 2016 Leeds, UK
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Talk type: Conference talk
What are variables of first-order logic and the lambda-calculus?
What a nice event the Logic Colloquium was. What a treat to be surrounded by people who care so much about logic I learned a lot.
Thanks to the Logic Colloquium.
Logic Colloquium 2016 Leeds, UK
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Talk type: Seminar
Nominal techniques and consistency of Quine's NF
Thanks to the LFCS Seminars series.
LFCS seminar series, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Nominal: the big picture
Thanks to the Birmingham Theory Seminars.
Birmingham Theory Seminar University of Birmingham, UK
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Talk type: Seminar
Nominal foundations of mathematics
Thanks to the Heriot-Watt Computer Science Seminar Series.
Computer Science Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Colloquium
Consistency of Quine's NF using nominal techniques
Thanks to the King’s College London Informatics Research Colloquium Series.
Informatics Research Colloquium, King's College London London, UK
2015
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Thanks to the Manchester Logic Seminar series.
Manchester Logic Seminar, University of Manchester Manchester, UK
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Thanks to the Logic and Semantics seminar series.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Teaching talk
On technical writing
A talk given to MSc students on (you guessed it) technical writing. We teach CS students to do technical things, but rarely do we focus on how to communicate about technical things. This can lead to problems later on, so I feel that this is an important lack and it is one which I am always happy to try and address.
Presented by Murdoch J. Gabbay MSc course, Heriot-Watt University Edinburgh, UK
2014
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Talk type: Seminar
Consistency of Quine’s NF using nominal techniques
Presented by Murdoch J. Gabbay Midlands Logic Seminar
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Talk type: Seminar
What sequent quantifier rules tell us about nominal semantics for logic
A precis of parts of nominal semantics for predicate logic and semantics out of context.
Thanks to the Leeds Logic Group.
Leeds Logic Group, University of Leeds Leeds, UK
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
See the announcement.
Thanks to the Leeds Logic Group.
Leeds Logic Group, University of Leeds Leeds, UK
2013
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Talk type: Workshop talk
Nominal duality theory for new, forall, and lambda
I was privileged to be invited to the rather wonderful Dagstuhl seminar 13422 on Nominal Computation Theory in Dagstuhl, Germany.
Thanks to the organisers for this great event. Shall we do another one?
Dagstuhl Seminar 13422: Nominal Computation Theory Schloss Dagstuhl, Germany
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Talk type: Seminar
A (partial) survey of nominal techniques
Just as the title says!
Thanks to Neil Ghani and his lively and interesting group for the very enjoyable afternoon.
University of Strathclyde Glasgow, UK
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Talk type: Seminar
An overview of nominal algebra, lattice, representation and dualities for computer science foundations
Presentation at the Logic and Semantics Seminar in the Computer Lab, Cambridge. I discuss how nominal algebra and powersets naturally model first-order logic with equality and the lambda-calculus. We can base theories of lattices, algebra, and topology on these observations, extending the familiar style of Boolean and Heyting algebras and their representations and topologies.
Thanks to Jonathan Hayman.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Seminar
Denotation of Contextual Modal Type Theory
Presentation at the Programming Research Group Seminar in the Computer Lab, Cambridge. See also the associated paper Denotation of … (a prize for the reader who guesses the rest of the title). Many thanks to Alan Mycroft and his team for being delightful and enthusiastic hosts.
Programming Research Group Seminar, University of Cambridge Cambridge, UK
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Talk type: Conference talk
Semantics of FOL and lambda-calculus using nominal techniques
Presentation at the Samson@60 conference in honour of Samson Abramsky. Getting to this event was unusually challenging and so I have an unusually long list of kind, generous, and efficient people to thank for my being able to attend: thanks to Luke Ong, Prakash Pananganden, Dave Corne, and Destiny Chen.
Samson@60 conference University of Oxford, UK
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Talk type: Seminar
An informal survey of nominal techniques
I sketch recent activity in nominal techniques and speculate on how nominal techniques could be useful in verification. The talk was aimed especially those not familiar with nominal techniques. I gave the talk without slides.
IMDEA Software Institute Madrid, Spain
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Talk type: Seminar
Denotation of Contextual Modal Type theory
Nanevski in his PhD thesis introduced and studied Contextual Modal Theory Theory. This is a lambda-calculus based via the Curry-Howard correspondence on an extension of the modal logic S4 with multiple modalities, one for each variable. The application is to meta-programming, where the intuitive denotation of Box A is “syntax of type A”. The language itself is pretty, and we developed a denotational semantics with interesting properties. I will give a tour of the system and discuss future work. See also the paper on Denotations of CMTT. Thanks to César Sánchez and Aleksandar Nanevski, and thanks to Paola Huerta for the calm and efficient rescue from that first hotel.
IMDEA Software Institute Madrid, Spain
2012
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Talk type: Workshop talk
Stone duality for first-order logic, a nominal approach
Presentation at the Workshop on Duality Theory in Algebra, Logic and Computer Science in the Mathematical Institute, Oxford. See also the associated paper Stone duality for first-order logic: a nominal approach to logic and topology. This was a particularly nice do, and a lovely chance for me to rub shoulders with a new community. One particular pleasure of the workshop was meeting a Polish academic who as a young man had studied with Mostowski. He told me some stories about the great man, and said how nice it was to see this mathematics being used.
Workshop on Duality Theory in Algebra, Logic and Computer Science Mathematical Institute, Oxford, UK
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Talk type: Seminar
Three semantics for logic out of context
It is common to reason about semantics in a context. This context may be a valuation (assigning denotations to variables) or a typing context (assigning types to variables), or a memory state (enumerating known memory), and so on. I will outline a nominal approach to semantics which subsumes context into a more abstract nominal notion of support. Using this we can give three semantics to (first-order) logic: in lattices, sets, and algebras. The full details are technical and in this talk I will try to bring out the non-technical “meaning of it all”, with an emphasis (necessarily speculative, but realistic) on practical implications to logic, specification, and programming.
Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Seminar
The algebraisation of logic using nominal techniques
Over the past five years a number of systems, such as first-order logic and the lambda-calculus, have been algebraically axiomatised using nominal techniques. A next step is to relate these (equational) axiomatisations to order-theoretic notions such as complemented bounded lattices (a.k.a. Boolean algebras). I will talk about recent progress here, giving an order-theoretic account of first-order logic: the punchline is that it will correspond to a complemented finitely fresh-complete lattice with a substitution—wait for the talk or see my recent paper with Dowek Nominal semantics for predicate logic: algebras, substitution, quantifiers, and limits. Israel has a tradition in logical foundations. Fraenkel–the Fraenkel of Zermelo-Fraenkel and Fraenkel-Mostowski set theory–was Dean of Jerusalem University. So it was special while visiting the logic community in Israel to discover not only that they already knew the background to what I was talking about, but that also, their research had anticipated much of what I was about to say. Special thanks here to Arnon Avron.
Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
The application of symmetry to computing and meta-mathematics
Symmetry has long been recognised as important in mathematics and physics, but its importance has been less appreciated in theories of logic and computation. In this (mathematically oriented, but non-technical) talk I describe what symmetry means in logic and computation, examine the benefits we can obtain by considering logic and computation as inherently “symmetric” mathematical entities, and relate this back to traditional foundations based on sets and functions. Special thanks here to Stuart Margolis and the remarkable Plotkin family.
Bar-Ilan University Tel Aviv, Israel
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Talk type: Conference talk
Nominal semantics for predicate logic
Presentation at the 9th Italian Convention on Computational Logic (CILC’2012) in Rome, Italy. See also the associated paper, which is called Nominal semantics for predicate logic.
9th Italian Convention on Computational Logic (CILC 2012) Rome, Italy
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Talk type: Seminar
You are not a number: how nameless foundations took over the world and made us think we have only numbers, and no names.
Seminar at the School of Natural and Computing Sciences at the University of Aberdeen. I describe the nature of and motivation for nominal research.
School of Natural and Computing Sciences, University of Aberdeen Aberdeen, UK
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Talk type: Workshop talk
On proof-search
Talk given at the mathematically structured programming group as part of the STP (Scottish Theorem Provers) meeting. I describe upcoming research to give a semantics to meta-variables in proof-search and theorem-proving.
Scottish Theorem Provers meeting, University of Strathclyde Glasgow, UK
2011
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Talk type: Workshop talk
Stone duality for first-order logic, a nominal approach
Given at the Howard Barringer workshop. Not a repeat of my talks at Paris or Edinburgh (see below), because I only had 20 minutes. Few technical details, much more discussion of the “conceptual stack” of nominal techniques and how I plan to develop the field. A very nice and interesting event with many varied speakers. The Festschrift is where the accompanying paper appears.
Howard Barringer workshop, University of Manchester Manchester, UK
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Talk type: Seminar
Stone duality for first-order logic, a nominal approach
A talk given at the work group on Types and Realisability / Théorie des types et réalisabilité”. A repeat of my talk of the previous week in Edinburgh, but to a different audience (and getting a different set of very constructive comments). Interesting to present the same work to two different audiences and see the responses.
Types and Realisability working group, INRIA Paris, France
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Talk type: Seminar
Stone duality for first-order logic, a nominal approach
Part of the LFCS seminar series in Edinburgh University. A brand-new duality result between appropriate nominal notions of topological space, and the nominal algebra axiomatisation of first-order logic from One-and-a-halfth order logic and Permissive-nominal logic.
LFCS seminar series, University of Edinburgh Edinburgh, UK
- Slides for Stone duality for first-order logic, a nominal approach
- Related paper for Stone duality for first-order logic, a nominal approach
- One-and-a-halfth-order logic for Stone duality for first-order logic, a nominal approach
- Permissive Nominal Logic for Stone duality for first-order logic, a nominal approach
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Talk type: Workshop talk
Getting to the Box of it: proof-theory for meta-programming and modal type theory
Talk given at the wonderful Roy Dyckhoff retirement event. I present new work with Aleksander Nanevski to give a denotational semantics to Contextual Modal Type Theory (CMTT).
Roy Dyckhoff retirement event University of St Andrews, UK
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Talk type: Seminar
Are all substitutions invertible; are all monoids groups?
Talk given as part of the theoretical computer science seminars discussing how to injectively turn an arbitrary monoid into a group. Joint work with Peter Kropholler.
Theoretical Computer Science Seminar, University of Birmingham Birmingham, UK
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Talk type: Workshop talk
Nominal Henkin semantics
Talk given at the 6th International Workshop on Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2011). Joint work with Dominic Mulligan.
6th International Workshop on Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2011) Nijmegen, Netherlands
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Talk type: Seminar
An overview of nominal techniques, semantics, and logic
Given at the Artificial Intelligence Research Institute (IIIA) in Barcelona, Spain. Does what it says on the tin. I gave the talk on the whiteboard; there are no slides.
Artificial Intelligence Research Institute (IIIA) Barcelona, Spain
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Talk type: Conference talk
Metamathematics based on nominal terms: first-order logics over nominal sets
A talk given at British Colloquium for Theoretical Computer Science (BCTCS 2011) in Birmingham, UK. I discuss the application of nominal sets to designing metamathematical syntaxes.
British Colloquium for Theoretical Computer Science (BCTCS 2011) Birmingham, UK
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Talk type: Workshop talk
Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
A talk given in the Automated Reasoning Workshop in Glasgow University, UK. I describe permissive nominal terms and their application in logic and algebra.
Automated Reasoning Workshop University of Glasgow, UK
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Talk type: Seminar
Nominal techniques: a success story of the continued relevance of mathematical foundations to computer science
A talk given at Glasgow University maths department. I discuss nominal techniques from a foundational point of view.
Mathematics Seminar, University of Glasgow Glasgow, UK
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Talk type: Seminar
A gentle introduction to the shape of nominal techniques
The same beginner-friendly talk I gave in February. I gave the talk on the whiteboard; there are no slides.
University of Strathclyde Glasgow, UK
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Talk type: Seminar
A gentle introduction to the shape of nominal techniques
A beginner-friendly talk given at the University of Glasgow, UK on nominal techniques. The abstract and announcement are here. I gave the talk on the whiteboard; there are no slides.
University of Glasgow Glasgow, UK
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Talk type: Seminar
Nominal techniques, 2006-2011
Given at Tel-Aviv university. I broadly survey of the development of nominal terms in logic since 2006 when I last visited the department, discussing permissive-nominal algebra and permissive-nominal logic. I gave the talk on the whiteboard; there are no slides.
Tel Aviv University Tel Aviv, Israel
2010
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Talk type: Workshop talk
Tattered type theories: type theories with holes
Part of the fun in the afternoon series of meetings. I describe current work with Dominic Mulligan on representing incompleteness in theorem-provers.
Thanks to Conor McBride.
Fun in the Afternoon meeting, University of Strathclyde Glasgow, UK
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Talk type: Seminar
Nominal techniques: logique, algèbre, calcul
A talk given at the 16th meeting on Logic, Algebra, and Calculus at PPS in Paris, France. I talk about applications of nominal rewriting, algebra, and permissive-nominal logic to specifying and reasoning about logic, algebra, and calculus.
Thanks to Delia Kesner and DIGITEO.
16th meeting on Logic, Algebra, and Calculus, PPS Paris, France
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Talk type: Seminar
Nominal sets and the NEW-quantifier
A talk given at the Typical Seminar at the École Polytechnique in Paris, France. I describe recent work on a representation theorem for the NEW-quantifier.
Thanks to Gilles Dowek and DIGITEO.
Typical Seminar, LIX, École Polytechnique Paris, France
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Talk type: Conference talk
Permissive nominal logic and all that
Given at the 2010 British Logic Colloquium in Birmingham, UK. I describe in broad terms names and their symmetry properties.
Thanks to Eike Ritter.
British Logic Colloquium 2010 Birmingham, UK
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Talk type: Workshop talk
Closed nominal rewriting and efficiently computable nominal algebra equality
A talk given at the 5th International Workshop on Logical Frameworks and Meta-languages (LFMTP 2010) in Edinburgh, UK. I discuss the connection between nominal rewriting and nominal algebra. Joint work with Maribel Fernández.
5th International Workshop on Logical Frameworks and Meta-languages (LFMTP 2010) Edinburgh, UK
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Talk type: Workshop talk
Kripke-style models in which logic and computation have equal standing
I discuss Kripke models of the untyped lambda-calculus at the lovely and exciting meeting in honour of Dana Scott.
Scott in Scotland meeting, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
PNL: Permissive-nominal logic
Given as part of the Computer Science Seminars in Leicester, UK. I discuss permissive-nominal logic; an extension of first-order logic with names and binding.
Thanks to Alexander Kurz.
Computer Science Seminar, University of Leicester Leicester, UK
2009
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Talk type: Workshop talk
Permissive nominal terms
A talk given at the Scottish Theorem Proving meeting in Scotland, UK. I discuss permissive nominal terms. I used the whiteboard; there are no slides.
Thanks Grant Passmore.
Scottish Theorem Proving meeting, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
Names: I denote, therefore I am
A talk given at the SICSA Complex Systems Engineering Joint THREADSS/SEMANTICS Meeting in Scotland, UK. I discuss in general (philosophical?) terms some of the ideas and motivations underlying my research.
Thanks to Phil Trinder.
SICSA Complex Systems Engineering Joint THREADSS/SEMANTICS Meeting Heriot-Watt University, Edinburgh, UK
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Talk type: Seminar
Permissive nominal terms
A 30 minute ‘theory lunch’ seminar given at IMDEA in Madrid, Spain. I discuss permissive nominal terms. I used the whiteboard; there are no slides.
Thanks to Alejandro Sanchez.
Theory Lunch, IMDEA Software Institute Madrid, Spain
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Talk type: Course / lecture
Nominal techniques: a theory of names and binding
A series of three 90 minute talks on nominal techniques given at the Universidad Complutense in Madrid, Spain. I used the whiteboard; there are no slides.
Thanks to Yolanda Ortega Mallen.
Universidad Complutense de Madrid Madrid, Spain
- Nominal techniques: a theory of names and binding 6–22 October 2009 (three 90-minute lectures; individual dates not recorded) Nominal-techniques lecture series Universidad Complutense de Madrid Madrid, Spain
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Talk type: Conference talk
Permissive nominal terms and their unification
A talk given at CILC09, the 24th Italian Conference on Computational Logic in Ferrara, Italy. See also the associated paper. I discuss permissive-nominal terms, which are similar to nominal terms but with properties more like those of first-order syntax. In particular, permissive-nominal terms lack freshness contexts, have inherent notions of alpha-equivalence and ‘free atoms of’, and the notion of solution of a unification or matching problem is based just on substitution.
24th Italian Conference on Computational Logic (CILC 2009) Ferrara, Italy
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Talk type: Workshop talk
Semantic Nominal Terms
A talk given at the 2nd International Workshop on Theory and Applications of Abstraction, Substitution and Naming in York, UK (a satellite event of ETAPS 2009). See also the associated paper. I discuss a semantics for nominal terms unknowns as ‘infinite streams of distinct atoms’. This reconciles atoms-abstraction with nominal terms. I also describe a form of substitution that is both capturing (as in nominal terms) and capture-avoiding (as in first-order syntax with binding) — at the same time.
2nd International Workshop on Theory and Applications of Abstraction, Substitution and Naming University of York, UK
2008
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Talk type: Course / lecture
The laws of writing a good dissertation
I discuss technical writing for an audience of fourth year undergraduates, on Wednesday 22 October at 13:15. Oh, what are the laws of writing a good dissertation? Read the slides and find out, with examples!
Heriot-Watt University Edinburgh, UK
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Talk type: Seminar
The nature of sets in computer science
A talk given at King’s College as part of their Advanced Programming Techniques seminar series, on Monday 29 September at 12:00. I show how from a belief in the empty set and in ordinals we can construct a mathematical universe large enough that mathematical objects (in particular, numbers, rationals, and reals) can be interpreted in it. In response to questions from the audience I discuss the difference between structure (which exists in the model) and meaning (which exists in our own minds). It was a pretty good audience!
Advanced Programming Techniques Seminar, King's College London London, UK
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Talk type: Seminar
Nominal terms and one-and-a-half level Curry Howard
A talk given at the Cambridge University computer lab as part of their Logic and Semantics seminar series, on Friday 26 September at 2pm. Joint work with Dominic Mulligan.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Workshop talk
Two-and-a-half order lambda-calculus
Talk given on Friday 4 July in the WFLP 2008 in Italy at 10:00. I describe two-and-a-half order lambda-calculus, a two-level lambda-calculus. Normally, the lambda-calculus has a capture-avoiding substitution. The two-level lambda-calculus is actually two lambda calculi, with distinct namespaces for variable symbols of `strong’ and `weak’ variables, but sharing a common application operator. Substitution for a strong variable does not avoid capture by lambda-abstraction for a weak variable. This leads to interesting and unusual phenomena.
Workshop on Functional and Logic Programming (WFLP 2008) Siena, Italy
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Talk type: Seminar
Nominal Algebra
Talk given on Monday 30 June in the Pisa University Department of Informatics Seminar series in Italy at 17:30. I describe nominal algebra, a nominal algebraic reasoning language.
Department of Informatics Seminar, Università di Pisa Pisa, Italy
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Talk type: Conference talk
Substitution for Fraenkel-Mostowski Foundations
Talk given on Friday 4 April in the AISB conference in Aberdeen at 17:30. I describe a substitution action defined for the atoms on Fraenkel-Mostowski sets (the general set theory version of nominal sets). In the original Gabbay-Pitts ‘nominal’ approach to modelling syntax, the atoms in Fraenkel-Mostowski set theory were used to model variable symbols. Since they have a substitution action, it may be that they can be used to model variables, as well. Joint work with Michael Gabbay. The associated paper is here.
AISB Convention University of Aberdeen, UK
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Talk type: Seminar
One-and-a-halfth order Curry Howard: Incomplete Derivations
A talk given at the École Polytechnique on Tuesday 4 March at 11am. Joint work with Dominic Mulligan.
École Polytechnique / LIX Paris, France
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Talk type: Workshop talk
Arbitrary objects in mathematics and semantics
Talk delivered by Michael Gabbay in the Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics on Thursday February 28. This talk was delivered by my co-author Michael Gabbay; a photo of Michael delivering the talk is online here. We connect relatively recent developments in Fraenkel-Mostowski sets and nominal style semantics, with existing philosophical thinking on arbitrary-ness and unknown entities in language, mathematics, and semantics. Joint work with Michael Gabbay. The associated paper is here.
Presented by Michael Gabbay Arché workshop on abstract objects in semantics and the philosophy of mathematics Paris, France
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Talk type: Seminar
Two-and-a-halfth order lambda-calculus: a calculus of the informal meta-level
A talk given in the DSG (Dependable Systems Group) meeting. Joint work with Dominic Mulligan.
Dependable Systems Group meeting, Heriot-Watt University Edinburgh, UK
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Talk type: Colloquium
Term sequent logics: cut your terms
Sequents, entailments, and cut-elimination — applied to term rewriting and reduction.
Presented by Michael Gabbay With Murdoch Gabbay Department of Computer Science colloquium King's College London, Department of Computer Science Hosted by King's College London London, UK
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Talk type: Colloquium
Semantics of names and binding
A King’s College London departmental colloquium given on January 14 at noon. I gave this talk the old-fashioned way on the blackboard; hence, no pdf slides online. It was good fun.
Thanks to Iman Poernomo.
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Talk type: Teaching talk
Strange but true! Second year undergraduates chat all night online about logic course
A talk given as part of the Flexible Learning Talks series in Heriot-Watt University on Friday 11 January. Thanks to Nicole Cargill-Kipar.
Flexible Learning Talks series, Heriot-Watt University Edinburgh, UK
2007
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Talk type: Seminar
Nominal techniques: present, and future prospects
Given in the Prose seminar series at TU/e in Eindhoven on November 14 at 15:30. I talk about where I think things should be going.
Prose seminar series, Eindhoven University of Technology Eindhoven, Netherlands
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Talk type: Workshop talk
The lambda context calculus
Given at LFMTP, part of CADE07 near Bremen on July 15 at 17:00. Joint work with Stéphane Lengrand”.
Logical Frameworks and Meta-languages: Theory and Practice (LFMTP), CADE 2007 Bremen, Germany
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Talk type: Workshop talk
a-logic with arrows
Workshop on Functional and Logic Programming, RDP 2007 Paris, France
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Talk type: Seminar
Resourceful truth: elementary models of resource-sensitive logics
A talk given at the LFCS logic and semantics club in Edinburgh University on June 15 at 13:00. Thanks to Bartek Klin.
Logic and Semantics Club, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Resourceful truth: elementary models of resource-sensitive logics
A talk given at Queen Mary and Westfield New College on June 4 at 15:30. Thanks to Hongseok Yang.
Queen Mary, University of London London, UK
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Talk type: Workshop talk
Names, computations, logics: the hole story.
Given at the Mathematical Theories of Abstraction, Substitution and Naming in Computer Science in ICMS, Edinburgh on May 27 at 12:05. Thanks to the organisers Fernández, Pitts, and Stark. I describe my current take on ‘nominal techniques’.
Mathematical Theories of Abstraction, Substitution and Naming in Computer Science ICMS, Edinburgh, UK
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Talk type: Teaching talk
Strange but true! 2nd year undergraduates chat all night online about logic course.
A talk given at the Flexible Learning Conference in Edinburgh at Heriot-Watt on May 18 at 14:05. I discuss my experience teaching the F22HO2 ‘formal specification’ course, and in particular the success of a message board I set up for the course, which attracted 1600 posts in five weeks and which remains active today. The video is the worst video in the history of worst videos. Sorry that I put the camera on the projector so it picked up the noise from the fan.
Flexible Learning Conference, Heriot-Watt University Edinburgh, UK
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Talk type: Seminar
Denotations for functional programs with names
I consider denotations for functional programming in substitution sets, an algebraic semantics for substitution.
Presented by Murdoch J. Gabbay DSG Digest, Heriot-Watt University Edinburgh, Scotland
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Talk type: Seminar
Fraenkel-Mostowski atoms model variables as well as names
A talk given at SPLS in Edinburgh on March 8 at 14:30. In this non-technical talk, we discuss a recent paper in which I exhibit a capture-avoiding substitution action on the Fraenkel-Mostowski sets universe. This suggests a new view of variables as denotational objects (names with a substitution action). Thanks to Kenneth MacKenzie.
Scottish Programming Languages Seminar, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Names and variables
A talk given at City University on February 27 at 12:00. We discuss our research intentions to investigate denotations in which names are first-class denotational citizens; why one might want to do this, and how we are doing it. Thanks to Artur Garcez.
City University London, UK
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Talk type: Seminar
Fraenkel-Mostowski atoms model variables as well as names
A talk given at Heriot-Watt University on February 20 at 14:00. We discuss a recent paper in which I exhibit a substitution action on the Fraenkel-Mostowski sets universe. Thanks to Pierluigi Frisco.
Heriot-Watt University Edinburgh, UK
2006
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Talk type: Seminar
Boxology
Given at St Andrew’s University on December 19 at 11:00. We discuss boxology and hierarchical nominal techniques. Thanks to Kevin Hammond.
University of St Andrews St Andrews, UK
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Talk type: Seminar
Nominal Algebra: a NEW mathematics of variables
A talk given at the LMU Institut für Informatik in Munich on December 6 at 16:00. We describe nominal algebra, its use to naturally represent meta-level reasoning on theories with binding, then concentrate on a nominal algebra axiomatisation of substitution and argue that ‘variable’ can be modelled as a ‘name plus substitution action’. Thanks to Martin Hofmann.
Institut für Informatik, LMU Munich Munich, Germany
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Talk type: Workshop talk
Substitution in Fraenkel-Mostowski sets
Given at the Centre for Mathematical Studies as part of the Cameleon meeting on December 2 at 16:00. We show how the model of names given by atoms in Fraenkel-Mostowski sets can be extended with a substitution action to a model of variables. Thanks to Thomas Forster.
Cameleon meeting, Centre for Mathematical Sciences, University of Cambridge Cambridge, UK
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Talk type: Seminar
Nominal Logic
A talk given at the University of Bath Computer Science Department on November 13 at 11:15. We discuss nominal logic and the slogans ‘logic with names’ versus ‘logic with variables’. Thanks to Mark Price and Alessio Guglielmi.
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Talk type: Seminar
Logic of names and freshness (splitting the atom)
A talk given at the Dependable Systems Group weekly seminar at Heriot-Watt on October 27 at 14:15. We discuss what happens to the lambda binder in the presence of a hierarchy of variables.
Dependable Systems Group Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
Nominal Algebra
Presented by my PhD student Aad Mathijssen in the 18th Nordic Workshop on Programming Theory (NWPT06) at Reykjavík University in Iceland. The corresponding abstract is here.
Presented by Aad Mathijssen 18th Nordic Workshop on Programming Theory Reykjavík University, Iceland
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Talk type: Workshop talk
Nominal techniques: evolving the atom
A talk given at the Scottish Theorem Proving meeting in the School of computer science. We give a position talk on nominal techniques considered useful as a means to generalise compositionality. Thanks to Phil Trinder and Rob Rothenberg.
Scottish Theorem Proving meeting, University of St Andrews St Andrews, UK
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Talk type: Teaching talk
The lambda-calculus
A brief talk-ette (talk-chen, talk-ino) presented to test my presentation skills as part of the PGCAP course.
PGCAP course, Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
Incomplete objects using the NEW calculus of contexts
Given at the DREAM workshop of the mathematical reasoning group at Edinburgh University on September 1. We speculate on higher-order logic in the NEW calculus of contexts.
DREAM workshop, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
Hierarchical nominal rewriting
Kindly presented on my behalf by James Cheney at LFMTP06 in Seattle, USA on Wednesday 16 August 2006 at 10:30.
Presented by James Cheney Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2006) Seattle, USA
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Talk type: Conference talk
One-and-a-halfth-order-logic
Presented at PPDP06 in Venice, Italy on Tuesday 11 July 2006 at 16:30.
Principles and Practice of Declarative Programming (PPDP 2006) Venice, Italy
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Talk type: Seminar
Concrete models of nominal algebra substitution
Presented at the CANS seminar on Thursday 22 June 2006 at 3pm. Thanks to Maribel Fernández.
CANS Seminar, King's College London London, UK
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Talk type: Seminar
Concrete models of nominal algebra substitution
Presented at the CISA/LFCS seminar at Edinburgh University on Tuesday 13 June 2006 at 4pm in room 4.03. Thanks to Lucas Dixon. (See also a revised and extended version.)
CISA/LFCS Seminar, University of Edinburgh Edinburgh, UK
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Talk type: Course / lecture
Fraenkel-Mostowski sets and the NEW model of abstraction
Presented as part of a course in Tel-Aviv University at 12:00 on Sunday 28 May.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Evolved logic: the next step
Presented at the programming languages group in Tel-Aviv University at 14:00 on Sunday 28 May. Thanks to Mooly Sagiv.
Programming Languages Group, Tel Aviv University Tel Aviv, Israel
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Talk type: Course / lecture
A NEW calculus of contexts
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 12:00 on Sunday 14 May. Also see gabbay:newcc.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Course / lecture
On substitution
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 10:30 on Thursday 11 May. Also see gabbay:capasn and gabbay:subaan.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Course / lecture
Nominal algebra with applications
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 10:30 on Thursday 4 May. Thanks to Nachum Dershowitz. Presented again at the Tel-Aviv University computer science departmental seminar at 10:00 on Tuesday 9 May. Only got half way through thanks to diligent questioning; I will deliver the second half at 10:00 on Tuesday 16 May. See papers gabbay:noma for the first half, and gabbay:onaah for the second.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Functional programming and unification with meta-variables
A research seminar given with the Babel group at the Universidad Politecnica de Madrid on March 31. We discuss our recent axiomatisation of substitution, and the NEW calculus of contexts, suggesting connections with logic programming and describing what we think might naturally be done next in this area. Thanks to Julio Marino.
Babel Group Seminar, Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Seminar
Nominal: an overview
Presented at the Departmental Seminar at the Universidad Politecnica de Madrid on March 31. We give an overview of (one large part of) our research, focussing on ‘nominal’. Thanks to Manuel Hermenegildo, Juan-Jose Moreno, and Jim Lipton for the invitation to visit.
Departmental Seminar, Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Workshop talk
A concrete model of linearity and separation
Presented at the Scottish Theorem Proving Meeting at Heriot-Watt University on March 17. We describe what we think is probably a uniform sets-based semantics for linear logic and bunched implications. Thanks to Andrew Ireland. I repeated this talk as a research seminar with the Babel group at the Universidad Politecnica de Madrid on March 28 with title “A concrete semantics for truth in bunched implications and linear logic?”. Thanks to Julio Marino.
Scottish Theorem Proving Meeting, Heriot-Watt University Edinburgh, UK
- A concrete model of linearity and separation Scottish Theorem Proving Meeting Heriot-Watt University Edinburgh, UK Slides for A concrete model of linearity and separation
- A concrete semantics for truth in bunched implications and linear logic? Babel group research seminar Universidad Politécnica de Madrid Madrid, Spain Slides for A concrete semantics for truth in bunched implications and linear logic?
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Talk type: Colloquium
Is substitution an abstract notion?
Departmental colloquium of March 13. We describe an axiomatisation of substitution in ‘nominal style’ and discuss its set-based semantics. Thanks to Iman Poernomo.
Departmental Colloquium, King's College London London, UK
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Talk type: Seminar
Separation
Departmental seminar given at Heriot-Watt on March 6. We report on joint work with Joe Wells in which we consider semantics for multiplicative and additive implication. Thanks to Lilia Georgieva.
Departmental Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Course / lecture
A NEW semantics of substitution
Report on joint research carried out with two students, Samuel Rota Bulo and Andrea Marin, in the framework of a course taught in the University of Venice. Presented by the students on March 1. Thanks to their tutor Simonetta Balsamo.
Presented by Samuel Rota Bulo and Andrea Marin University of Venice Venice, Italy
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Talk type: Seminar
The NEW calculus of contexts
Talk given at the Scottish Programming Languages Seminar on February 1. We describe the NEW calculus of contexts, a lambda-calculus which tries to understand the meta-level/object-level split. Thanks to Edwin Brady. (I gave similar talks in Turin, Italy and at PPDP05.)
Scottish Programming Languages Seminar, University of St Andrews St Andrews, UK
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Talk type: Seminar
Nominal techniques and the meta-level
Talk given at Venice University on January 24. In reaction to the question “So what is it that you do?”, I give an initially non-technical description of the motivations (as I see them) for that part of my work to date which has to do with nominal techniques. Thanks to Annalisa Bossi.
University of Venice Venice, Italy
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Talk type: Colloquium
Nominal Rewriting
Talk given at the computational logic colloquium at Innsbruck University on January 23. We describe nominal rewriting as a meta-level framework for specifying rewriting, which accurately reflects the instantiation behaviour of meta-level variables with respect to object-level binding. We then expatiate way over the time limit on the general context of our research. Thanks to Aart Middeldorp for inviting me and to Georg Moser for making it happen.
Computational Logic Colloquium, University of Innsbruck Innsbruck, Austria
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Talk type: Workshop talk
The NEW calculus of contexts
Talk given at the Turin workshop on implicit computational complexity on January 19. We describe the NEW calculus of contexts, a lambda-calculus which tries to understand the meta-level/object-level split. Thanks to Simona Ronchi della Rocca. (A corrected and slightly modified version of a talk given at PPDP05.)
Workshop on Implicit Computational Complexity, University of Turin Turin, Italy
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Talk type: Seminar
The Fraenkel-Mostowski (FM) model of abstraction applied to name-generation
Given at the ULTRA group seminar at Heriot-Watt on January 9. We discuss the FM model of abstraction, and briefly suggest its usefulness for both inductive and coinductive types. Thanks to Lilia Georgieva.
ULTRA Group Seminar, Heriot-Watt University Edinburgh, UK
2005
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Talk type: Workshop talk
Automating first-order logic
Given at the second CANS meeting on December 6. We present Nominal Algebraic Specifications, a theory of universal algebra based on Nominal Terms and thus able to algebraically specify, in a suitable sense, theories-with-binding. We show how an algebraic theory of first-order logic leads to “one-and-a-halfth-order logic”, which is above first-order logic and slightly to the side of second-order logic. We discuss proof-methods for the system and possible extensions. Thanks to Maribel Fernández.
Second CANS meeting, King's College London London, UK
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Talk type: Colloquium
One-and-a-halfth-order logic
Given at the Institute for Logic, Language, and Computation, in their colloquium on mathematical logic. We present one-and-a-halfth-order logic, which is a logic somewhere above first-order logic and slightly to the side of second-order logic. It satisfies cut-elimination, has an algebraic presentation, and allows to quantify over all predicates in an interesting and unusual way. Thanks to Benedikt Loewe.
Mathematical Logic Colloquium, Institute for Logic, Language and Computation Utrecht University, Netherlands
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Talk type: Seminar
Nominal Logic and FM techniques: reasoning about binding without pain
Given at the Foundations Group seminar of the Brouwer Institute in Radboud University in Nijmegen on November 29. We give a relatively technical sketch of FM techniques, Nominal Logic, and their application to reasoning on syntax-with-binding. Thanks to Henk Barendregt and Bas Spitters. I gave the same talk at the Logic and Semantics Group Joint Theory seminar (‘Joint’ between Queen Mary and Westfield College and Imperial College) at Queen Mary and Westfield College on December 7 under the name “Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction”. Thanks to Peter O’Hearn. One member of the audience in Nijmegen then kindly invited me to give the talk again at the Theoretical Computer Science department in the Vrije Universiteit Amsterdam on December 12 (I used the original title). Thanks to Jan-Willem Klop and Femke van Raamsdonk.
Foundations Group Seminar, Brouwer Institute Radboud University, Nijmegen, Netherlands
- Nominal Logic and FM techniques: reasoning about binding without pain Foundations Group seminar, Brouwer Institute Radboud University Nijmegen, Netherlands Slides for Nominal Logic and FM techniques: reasoning about binding without pain
- Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction Logic and Semantics Group Joint Theory seminar Queen Mary and Westfield College London, UK Slides for Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction
- Nominal Logic and FM techniques: reasoning about binding without pain Theoretical Computer Science seminar Vrije Universiteit Amsterdam Amsterdam, Netherlands Slides for Nominal Logic and FM techniques: reasoning about binding without pain
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Talk type: Seminar
a-logic
Talk given in the Logic Tea series at the University of Amsterdam on November 22. We discuss a-logic, a first-order logic with a predicate atom which is true of variable symbols and false of non-variable symbol terms. We axiomatise the untyped lambda-calculus by way of example and discuss what models look like, some possible future work, and the design decisions (and some of the sins) hidden in the paper. Thanks to Johan van Benthem and Olivier Roy.
Logic Tea, University of Amsterdam Amsterdam, Netherlands
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Talk type: Seminar
Nominal algebraic specifications
Talk given at London South Bank University on November 8 and at University College London on November 9. We discuss Nominal Algebraic Specifications, an algebraic framework tailored to axiomatising systems-with-binding such as the lambda-calculus and logic. Thanks to Jonathan Bowen and Robin Hirsch.
London South Bank University London, UK
- Nominal algebraic specifications Research seminar London South Bank University London, UK Slides for Nominal algebraic specifications
- Nominal algebraic specifications Research seminar University College London London, UK Slides for Nominal algebraic specifications
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Talk type: Workshop talk
Nominal terms with a hierarchy of variables
Bleeding-edge talk given in Eindhoven in a group workshop on 31 October 2005. We give a speculative account of an algebraic version of the NEW calculus of contexts.
Group workshop, Eindhoven University of Technology Eindhoven, Netherlands
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Talk type: Seminar
Nominal Rewriting. Or ... talking about functions, without functions
Talk given at Tel-Aviv University on 6 October 2005. We give a non-technical account of Nominal Rewriting. Thanks to Arnon Avron and Nachum Dershowitz.
Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Nominal Rewriting
Talk given at Leicester University on 9 August 2005. We discuss Nominal Rewriting and suggest why it and the methods underlying it are of interest.
University of Leicester Leicester, UK
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Talk type: Seminar
a-logic, the lambda-calculus, and internalising meta using names and binding
Talk given at LMU in Munich on 26 July 2005. We give a broad view of our research, mentioning a-logic and a whole load of other stuff.
LMU Munich Munich, Germany
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Talk type: Seminar
Restart: a natural rule
Talk given at South Bank University in London on 25 July 2005. We discuss Restart, a Natural Deduction rule which turns intuitionistic logic into classical logic, without affecting proof-normalisation.
London South Bank University London, UK
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Talk type: Conference talk
Extended Nominal Rewriting: two models of binding
Talk given at PPDP05 in Lisbon on 10 July 2005. We discuss two notions of binding which exist side-by-side in Extended Nominal Rewriting; one modelling lambda-calculus-like abstraction, the other modelling pi-calculus-like restriction. Some typos fixed with respect to the similar talk in Paris, and only 30 minutes long.
Principles and Practice of Declarative Programming (PPDP 2005) Lisbon, Portugal
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Talk type: Conference talk
A NEW calculus of contexts
Talk given at PPDP05 in Lisbon on 10 July 2005. We discuss a lambda-calculus with a hierarchy of variables of different ‘strengths’ (corresponding roughly to increasingly ‘meta’ variables). We find it to be remarkably expressive, and suggest that we can encode records, modules, objects, dynamic binding and rebinding, and many other constructs involving non-capture-avoiding substitution.
Principles and Practice of Declarative Programming (PPDP 2005) Lisbon, Portugal
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Talk type: Seminar
Extended Nominal Rewriting: two models of binding
Talk given at LIX, Ecole Polytechnique, France, on 24 May. Thanks to Dale Miller. We discuss two notions of binding which exist side-by-side in Extended Nominal Rewriting; one modelling lambda-calculus-like abstraction, the other modelling pi-calculus-like restriction.
LIX, École Polytechnique Paris, France
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Talk type: Workshop talk
Is there a quick way to get names and binding in some nominal sense up and running in Isabelle/HOL polymorphically on the type-class ‘type’, without having to rewrite the whole thing from scratch — yes or no?
Talk given at the Binding Challenges workshop in JAIST, Ishikawa, Japan. Thanks to Rene Vestergaard. We speculate on an interesting way in which it may be possible to encode syntax-with-binding in a higher-order logic based theorem-proving environment.
Binding Challenges workshop, Japan Advanced Institute of Science and Technology Ishikawa, Japan
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Talk type: Seminar
Aspects of Nominal Rewriting
Slides of a talk given at the CWI in Amsterdam on 2 March, and TUE in Eindhoven on 3 March. Thanks to Anton Wijs and MohammadReza Mousavi.
CWI Amsterdam, Netherlands
- Aspects of Nominal Rewriting Research seminar Centrum Wiskunde & Informatica (CWI) Amsterdam, Netherlands Slides for Aspects of Nominal Rewriting
- Aspects of Nominal Rewriting Research seminar Eindhoven University of Technology Eindhoven, Netherlands Slides for Aspects of Nominal Rewriting
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Talk type: Seminar
A NEW calculus of contexts
Slides of a talk given in INRIA Rocquencourt Paris, on 21 February, and PPS Paris, on 22 February. Thanks to James Leifer, Russ Harmer, and the people of PPS and INRIA.
INRIA Rocquencourt Paris, France
- A NEW calculus of contexts Research seminar INRIA Rocquencourt Paris, France Slides for A NEW calculus of contexts
- A NEW calculus of contexts Research seminar PPS Paris, France Slides for A NEW calculus of contexts
2004
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Talk type: Colloquium
One quantifier for forall and exists and generic elements or ... unknowns are data
Slides of a talk given at the departmental colloquium on 24 November 2004.
Departmental Colloquium, King's College London London, UK
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Talk type: Seminar
Nominal Unification
Slides of a talk given in Imperial College, London, on 10 September 2004. Thanks to Stephen Muggleton.
Imperial College London London, UK
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Talk type: Conference talk
Nominal Rewriting Systems
Slides of a talk at PPDP04 on joint work with Maribel Fernández and Ian Mackie, given in Verona, Italy, on 26 August 2004. See also the paper.
Principles and Practice of Declarative Programming (PPDP 2004) Verona, Italy
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Talk type: Conference talk
A Sequent Calculus for Nominal Logic
Slides of a talk at LICS04 on joint work with James Cheney, given in Turku, Finland, on 15 July 2004. See also the paper.
Logic in Computer Science (LICS 2004) Turku, Finland
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Talk type: Seminar
Fresh Logic
We discuss Fresh Logic (joint work with James Cheney) and consider its relation to FOLN a logic by Miller and Tiu, especially in the light of insights from implementation. Talk given to the PARCIFAL group at LIX on 14 May 2004.
PARCIFAL Group, LIX Paris, France
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Talk type: Seminar
Nominal Terms, Existential Variables, and Mathematics
Talk given at PPS on 4 May 2004.
PPS Paris, France
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Talk type: Research talk
Extensions of Nominal Terms
We discuss extensions of Nominal Terms (see the paper on Nominal Unification). Lots of blue sky, but pretty.
Not recorded in the legacy archive
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Talk type: Seminar
Nominal Rewriting
We present Nominal Rewriting, an essentially first-order rewriting system with inbuilt support for alpha-equivalence. See also “Incomplete lambda-terms” below. Talk given in LRI in their seminar series (27/2/2004, 10:30am).
LRI Seminar France
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Talk type: Seminar
Incomplete lambda-terms
We use Nominal Rewriting to specify and discuss a lambda-calculus with meta-variables and explicit substitution. Talk given in LIX as part of a series of informal talks on meta-variables.
LIX Paris, France
2003
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Talk type: Workshop talk
Restart as a Computational Rule
Talk given on applying the Curry-Howard correspondence to a novel Restart Logic. Given in the Workshop on Lambda Calculus, Type Theory, and Natural Language in King’s College London, on 9 December at 12:00 on work.
Workshop on Lambda Calculus, Type Theory, and Natural Language King's College London, UK
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Talk type: Seminar
Fresh Logic: Vanilla First-Order Logic with FM?
Talk given on Fresh Logic (a first-order logic for the NEW-quantifier with proof-normalisation) on 2 June at the local group’s weekly Semantics Lunch. See also an abstract.
Semantics Lunch, University of Cambridge Cambridge, UK
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Talk type: Seminar
FM binding for pi-calculus transitions
Talk given on May 30 at Fundacao da Faculdade de Ciencias e Tecnologia / University of Lisbon. Thanks to Luis Caires and his department. See also an abstract.
University of Lisbon Lisbon, Portugal
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Talk type: Seminar
FreshML: programming with binders made easy
Talk given on May 28 at Fundacao da Faculdade de Ciencias e Tecnologia / University of Lisbon. Thanks to Luis Caires and his department. See also an abstract.
University of Lisbon Lisbon, Portugal
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Talk type: Seminar
FM and the pi-calculus
I investigate what happens if we consider an FM-like category of objects with a finite name-for-name renaming action (possibly non-injective maps between names) rather than a finite name-for-name permutation action as is the case for FM sets. I obtain a category similar to Schan, but which supports a natural interpretation of weak HOAS. Talk given on March 20 at the ENS Lyon, France. Thanks to Daniel Hirschkoff. See also an abstract.
École Normale Supérieure de Lyon Lyon, France
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Talk type: Seminar
FM for Higher-Order Abstract Syntax
I investigate what happens if we consider an FM-like category of objects with a finite name-for-name renaming action (possibly non-injective maps between names) rather than a finite name-for-name permutation action as is the case for FM sets. I obtain a category similar to Schan, but which supports a natural interpretation of weak HOAS. Talk given on March 13 in Manchester University, thanks to Peter Aczel. See also an abstract.
University of Manchester Manchester, UK
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Talk type: Seminar
Restriction, binding, and three presentations of the pi-calculus
I’ve been using FreshML to program a pet interest of mine; transition systems and bisimulation checkers for the pi-calculus (see [gabbay:thempc]). I’ll show excerpts from the programs I’ve written and talk about what writing them has taught me about another interest of mine; finding a semantics for restriction as distinct from binding. The sourcecode discussed is the pi-ltsb series of programs above, see [pi-ltsb] above.
Semantics Lunch, University of Cambridge Cambridge, UK
2002
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Talk type: Seminar
FM for Names and Binding
Talk given on 11/12/2002 at the Japan Advanced Institute of Science and Technology, Ishikawa, on 14/12/2002 at MLG 2002 in Kinosaki, and on 16/12/2002 at ALGI in Kyoto. Thanks to René Vestergaard.
Japan Advanced Institute of Science and Technology Ishikawa, Japan
- FM for Names and Binding Research seminar Japan Advanced Institute of Science and Technology Ishikawa, Japan Slides for FM for Names and Binding
- FM for Names and Binding MLG 2002 Kinosaki, Japan Slides for FM for Names and Binding
- FM for Names and Binding ALGI Kyoto, Japan Slides for FM for Names and Binding
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Talk type: Seminar
Models of pi-calculus behaviour using FM techniques
Talk given in Pisa on 2/12/2002. Thanks to Ugo Montanari.
University of Pisa Pisa, Italy
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Talk type: Seminar
Theory and Models of the pi-calculus using Fraenkel-Mostowski Generalised
Talk given in Cambridge on 11/11/2002 at the local group’s “Semantics Lunch”.
Semantics Lunch, University of Cambridge Cambridge, UK
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at the Max-Planck Institute, Saarbrücken, Germany.
Max Planck Institute Saarbrücken, Germany
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at INRIA Rocquencourt France 11/10/02 and Université Paris VII France 14/10/02. Thanks to Gilles Dowek and Pierre-Louis Curien.
INRIA Rocquencourt and Université Paris VII Paris, France
- FM Techniques for Syntax-with-Binding Research seminar INRIA Rocquencourt Paris, France Slides for FM Techniques for Syntax-with-Binding
- FM Techniques for Syntax-with-Binding Research seminar Université Paris VII Paris, France Slides for FM Techniques for Syntax-with-Binding
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at IBM Research, Haifa, Israel 2/10/02. Thanks to Cindy Eisner.
IBM Research Haifa, Israel
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Talk type: Workshop talk
FM-HOL
Contributed talk at Workshop on Thirty Five years of Automath, Informal Proceedings, Heriot-Watt University, Edinburgh, Scotland, April 2002.
Workshop on Thirty Five Years of Automath Heriot-Watt University, Edinburgh, UK
Talks by topic
Some talks appear under more than one topic.
Algebra 9 talks
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Talk type: Seminar
Imaginary groups
I visit the maths department.
Thanks to the Maximals seminar series.
Maximals seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Seminar
Are all substitutions invertible; are all monoids groups?
Talk given as part of the theoretical computer science seminars discussing how to injectively turn an arbitrary monoid into a group. Joint work with Peter Kropholler.
Theoretical Computer Science Seminar, University of Birmingham Birmingham, UK
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Talk type: Seminar
Nominal Algebra
Talk given on Monday 30 June in the Pisa University Department of Informatics Seminar series in Italy at 17:30. I describe nominal algebra, a nominal algebraic reasoning language.
Department of Informatics Seminar, Università di Pisa Pisa, Italy
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Talk type: Seminar
Nominal Algebra: a NEW mathematics of variables
A talk given at the LMU Institut für Informatik in Munich on December 6 at 16:00. We describe nominal algebra, its use to naturally represent meta-level reasoning on theories with binding, then concentrate on a nominal algebra axiomatisation of substitution and argue that ‘variable’ can be modelled as a ‘name plus substitution action’. Thanks to Martin Hofmann.
Institut für Informatik, LMU Munich Munich, Germany
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Talk type: Workshop talk
Nominal Algebra
Presented by my PhD student Aad Mathijssen in the 18th Nordic Workshop on Programming Theory (NWPT06) at Reykjavík University in Iceland. The corresponding abstract is here.
Presented by Aad Mathijssen 18th Nordic Workshop on Programming Theory Reykjavík University, Iceland
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Talk type: Seminar
Concrete models of nominal algebra substitution
Presented at the CANS seminar on Thursday 22 June 2006 at 3pm. Thanks to Maribel Fernández.
CANS Seminar, King's College London London, UK
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Talk type: Seminar
Concrete models of nominal algebra substitution
Presented at the CISA/LFCS seminar at Edinburgh University on Tuesday 13 June 2006 at 4pm in room 4.03. Thanks to Lucas Dixon. (See also a revised and extended version.)
CISA/LFCS Seminar, University of Edinburgh Edinburgh, UK
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Talk type: Course / lecture
Nominal algebra with applications
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 10:30 on Thursday 4 May. Thanks to Nachum Dershowitz. Presented again at the Tel-Aviv University computer science departmental seminar at 10:00 on Tuesday 9 May. Only got half way through thanks to diligent questioning; I will deliver the second half at 10:00 on Tuesday 16 May. See papers gabbay:noma for the first half, and gabbay:onaah for the second.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Nominal algebraic specifications
Talk given at London South Bank University on November 8 and at University College London on November 9. We discuss Nominal Algebraic Specifications, an algebraic framework tailored to axiomatising systems-with-binding such as the lambda-calculus and logic. Thanks to Jonathan Bowen and Robin Hirsch.
London South Bank University London, UK
- Nominal algebraic specifications Research seminar London South Bank University London, UK Slides for Nominal algebraic specifications
- Nominal algebraic specifications Research seminar University College London London, UK Slides for Nominal algebraic specifications
Blockchain 4 talks
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Talk type: Tech talk
Karl Marx and the blockchain
This talk is based on a book chapter on Karl Marx and the blockchain coauthored with Devraj Basu (which at least one reviewer really liked) and an article in the Times Higher Education. Marx was an economic thinker and a revolutionary, and it’s his economic thought that the talk and associated article focus on. The article unpacks a case for the relevance of Marxian economic thought to blockchain technology. Writing this was fun, and it was a nice change from pure mathematics.
Presented by Murdoch J. Gabbay Huawei Tech Talk Edinburgh, UK
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Talk type: Colloquium
A broadly Marxian analysis of blockchain and cryptocurrencies
This talk is based on a book chapter on Karl Marx and the blockchain coauthored with Devraj Basu (which at least one reviewer really liked) and an article in the Times Higher Education. Marx was an economic thinker and a revolutionary, and it’s his economic thought that the talk and associated article focus on. The article unpacks a case for the relevance of Marxian economic thought to blockchain technology. Writing this was fun, and it was a nice change from pure mathematics.
Presented by Murdoch J. Gabbay Middlesex Computer Science Colloquium
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Talk type: Tech talk
What is an EUTxO blockchain?
A revised and expanded exploration of a new and simple mathematical model of blockchains
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Seminar
What is an EUTxO blockchain?
An exploration of a new and simple mathematical model of blockchains
Thanks to Ohad Kammar. See the LFCS Seminar series videos online.
LFCS seminar series, University of Edinburgh Edinburgh, UK / online
Consensus and distributed systems 17 talks
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Talk type: Course / lecture
Distributed protocols: a logical approach
Distributed protocols are famously hard to get right, specify, and reason about. Logic can help! For example, it can help us give a three-day summer school course on a message-free, time-free, failure-free logical abstraction for specifying message-passing timed protocols with failure.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2026 (SPLV26) University of Glasgow Glasgow, Scotland
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Talk type: Research talk
Exponential cake and consistent controllers
This talk treats the consistent controllers problem: suppose you have multiple interacting state machines (controllers) A, B, C, and D. Suppose controller B is bad and might equivocate, meaning that it may report different and incompatible states to peers A, C, and D. How can peers A, C, and D detect this misbehaviour and gracefully recover to a consistent state? Answers that invoke a single controlling peer G that suspends A, C, and D and resets them to a most recent consistent state are not admissible; there is no deus ex machina here, just controllers trying to stay consistent. If this problem seems hypothetical – it’s not. The world already consists of interacting controllers and the problem of consistency is a serious brake on safe progress.
Presented by Murdoch J. Gabbay With Isaac Sheff, Anthony Hart Heliax Research Talk
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Talk type: Workshop talk
Heterogeneous Broadcast
A protocol is heterogeneous when different controllers may have different trust assumptions. A controller is a machine that issues state: a bank, a national register, a blockchain, etc. The world has many controllers, each of which a) have their own trust assumptions and b) must interoperate with the others. Multiple trust assumptions takes us to a distinct design space, so that Heterogeneous Bracha Broadcast is not merely a generalisation of (Homogeneous) Bracha Broadcast: it’s a different beast.
Presented by Murdoch J. Gabbay With Isaac Sheff Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
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Talk type: Workshop talk
Declarative Bracha Broadcast using 3-valued logic
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour. The result is simple and highly effective.
Presented by Murdoch J. Gabbay Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
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Talk type: Seminar
Declarative distributed protocols using three-valued modal logic and semitopologies
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour, and semitopologies provide a language for describing quorums. The result is to exhibit protocols as simple and compact axiomatic theories.
Presented by Murdoch J. Gabbay Paris Automata and Concurrency Theory Seminar Paris, France
- Slides for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Bracha Broadcast for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Paxos for Declarative distributed protocols using three-valued modal logic and semitopologies
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Talk type: Workshop talk
Declarative distributed protocols using three-valued modal logic types
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 26071: Behavioural Types for Resilience Schloss Dagstuhl Dagstuhl, Germany
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Against Cryptoeconomics Day
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Anoma Day
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Talk type: Conference talk
A declarative approach to specifying distributed algorithms using three-valued modal logic
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Workshop talk
Message logic for service commitments
Message logic is a logic for specifying and reasoning about promises that participants in a message-passing distributed system can make to one another. This sounds simple but it turns out to lead to a surprisingly nice logical framework. This logic is somewhat different from the logics I’m used to studying, notably because the notion of logical implication has a measure-theoretic flavour. This comes from the fact that promises made might be broken, even in good faith, e.g. if the entire network fails.
Presented by Murdoch J. Gabbay IC3 Winter Retreat
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Talk type: Workshop talk
Semitopologies and semiframes for distributed heterogeneous consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). Semitopologies naturally generate a three-valued logic (by considering continuous functions out of them), and they can be dualised to semiframes, which are complete join-semilattices with a compatibility relation. We study these, along with various well-behavedness conditions on points in a semitopology.
Presented by Murdoch J. Gabbay First Workshop on Heterogeneous Trust in Distributed Systems (HTDS 2023) Convocation Room 113, Friend Center Princeton, New Jersey, USA
- Slides for Semitopologies and semiframes for distributed heterogeneous consensus
- Recording of Semitopologies and semiframes for distributed heterogeneous consensus
- Related paper for Semitopologies and semiframes for distributed heterogeneous consensus
- Semiframes paper for Semitopologies and semiframes for distributed heterogeneous consensus
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Talk type: Workshop talk
Semitopology and (paraconsistent) logic
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). It turns out that semitopologies are most easily classified using a three-valued paraconsistent logic; put another way, when you try to develop a logic over semitopologies, with semantics as continuous functions, a third truth-value very naturally appears.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 23272: Epistemic and Topological Reasoning in Distributed Systems Schloss Dagstuhl Dagstuhl, Germany
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Talk type: Seminar
Semitopology: a new topological model of heterogeneous consensus
A quorum in a distributed system is a set of participants with sufficient power to take collective action, independently from the rest of the system. Quorums are closed under arbitrary union, but not under arbitrary intersection. It turns out that if you treat this as something that is nearly (but not quite) a point-set topology, and then replay topological notions like open set, closed set, dense set, and continuous function, interpreted as quorums, then you obtain constructs with real meaning in the theory of distributed systems (e.g. a dense set corresponds to a blocking set). It turns out further that if you take notions from distributed systems and view them topologically, then further rich structure is revealed.
Presented by Murdoch J. Gabbay Scottish Programming Languages Seminar (SPLS) Coats Building, University of the West of Scotland Paisley, Scotland
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Talk type: Seminar
The semitopology of heterogeneous consensus
A mathematical notion of consensus in a heterogeneous permissionless environment. ‘Permissionless’ means that participants are free to join or leave the network without permission from a central authority; ‘heterogeneous’ means that participants may have different goals and beliefs.
Thanks to the Online Social Choice and Welfare seminar series (permalink). Video online here.
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Talk type: Conference talk
The (semi)topology of distributed consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open. This generalises the notion of quorum from distributed systems. It soon becomes clear that the most interesting semitopologies are those that satisfy an intersection-of-open-sets property which is the precise negation of Hausdorff separation.
Presented by Murdoch J. Gabbay 11th International Conference on Geometric and Topological Methods in Computer Science (GETCO 2022) EPITA School of Engineering and Computer Science Le Kremlin-Bicêtre, France / hybrid
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Talk type: Tech talk
A semi-topological view of real-world consensus
Consensus = Topology
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Workshop talk
A semi-topological view of real-world consensus
Consensus = Topology Thanks to the FRIDA organisers.
7th Workshop on Formal Reasoning in Distributed Algorithms (FRIDA 2020) Online
Declarative distributed protocols 9 talks
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Talk type: Course / lecture
Distributed protocols: a logical approach
Distributed protocols are famously hard to get right, specify, and reason about. Logic can help! For example, it can help us give a three-day summer school course on a message-free, time-free, failure-free logical abstraction for specifying message-passing timed protocols with failure.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2026 (SPLV26) University of Glasgow Glasgow, Scotland
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Talk type: Research talk
Exponential cake and consistent controllers
This talk treats the consistent controllers problem: suppose you have multiple interacting state machines (controllers) A, B, C, and D. Suppose controller B is bad and might equivocate, meaning that it may report different and incompatible states to peers A, C, and D. How can peers A, C, and D detect this misbehaviour and gracefully recover to a consistent state? Answers that invoke a single controlling peer G that suspends A, C, and D and resets them to a most recent consistent state are not admissible; there is no deus ex machina here, just controllers trying to stay consistent. If this problem seems hypothetical – it’s not. The world already consists of interacting controllers and the problem of consistency is a serious brake on safe progress.
Presented by Murdoch J. Gabbay With Isaac Sheff, Anthony Hart Heliax Research Talk
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Talk type: Workshop talk
Heterogeneous Broadcast
A protocol is heterogeneous when different controllers may have different trust assumptions. A controller is a machine that issues state: a bank, a national register, a blockchain, etc. The world has many controllers, each of which a) have their own trust assumptions and b) must interoperate with the others. Multiple trust assumptions takes us to a distinct design space, so that Heterogeneous Bracha Broadcast is not merely a generalisation of (Homogeneous) Bracha Broadcast: it’s a different beast.
Presented by Murdoch J. Gabbay With Isaac Sheff Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
-
Talk type: Workshop talk
Declarative Bracha Broadcast using 3-valued logic
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour. The result is simple and highly effective.
Presented by Murdoch J. Gabbay Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
-
Talk type: Seminar
Declarative distributed protocols using three-valued modal logic and semitopologies
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour, and semitopologies provide a language for describing quorums. The result is to exhibit protocols as simple and compact axiomatic theories.
Presented by Murdoch J. Gabbay Paris Automata and Concurrency Theory Seminar Paris, France
- Slides for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Bracha Broadcast for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Paxos for Declarative distributed protocols using three-valued modal logic and semitopologies
-
Talk type: Workshop talk
Declarative distributed protocols using three-valued modal logic types
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 26071: Behavioural Types for Resilience Schloss Dagstuhl Dagstuhl, Germany
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Against Cryptoeconomics Day
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Talk type: Research talk
Heterogeneous Broadcast
I present Heterogeneous Bracha Broadcast; a broadcast protocol in which participants may have different trust assumptions (i.e. trust different sets of quorums). The name might suggest that Heterogeneous Bracha Broadcast is a generalisation of Bracha Broadcast, but it’s not really: introducing multiple quorum systems takes us into a new design space which (at least in this case) feels quite unlike the homogeneous case.
Presented by Murdoch J. Gabbay With Isaac Sheff Anoma Day
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Talk type: Conference talk
A declarative approach to specifying distributed algorithms using three-valued modal logic
I discuss how to specify distributed protocols as axiomatic theories in three-valued modal logic. As usual for axiomatic (‘declarative’) specifications, this yields a compact, precise, easily-checkable summary of what the protocol is, while dispensing with algorithmic detail.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
Lambda calculus and metaprogramming 16 talks
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Talk type: Seminar
Denotation of Contextual Modal Type Theory
Presentation at the Programming Research Group Seminar in the Computer Lab, Cambridge. See also the associated paper Denotation of … (a prize for the reader who guesses the rest of the title). Many thanks to Alan Mycroft and his team for being delightful and enthusiastic hosts.
Programming Research Group Seminar, University of Cambridge Cambridge, UK
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Talk type: Seminar
Denotation of Contextual Modal Type theory
Nanevski in his PhD thesis introduced and studied Contextual Modal Theory Theory. This is a lambda-calculus based via the Curry-Howard correspondence on an extension of the modal logic S4 with multiple modalities, one for each variable. The application is to meta-programming, where the intuitive denotation of Box A is “syntax of type A”. The language itself is pretty, and we developed a denotational semantics with interesting properties. I will give a tour of the system and discuss future work. See also the paper on Denotations of CMTT. Thanks to César Sánchez and Aleksandar Nanevski, and thanks to Paola Huerta for the calm and efficient rescue from that first hotel.
IMDEA Software Institute Madrid, Spain
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Talk type: Workshop talk
Getting to the Box of it: proof-theory for meta-programming and modal type theory
Talk given at the wonderful Roy Dyckhoff retirement event. I present new work with Aleksander Nanevski to give a denotational semantics to Contextual Modal Type Theory (CMTT).
Roy Dyckhoff retirement event University of St Andrews, UK
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Talk type: Workshop talk
Nominal Henkin semantics
Talk given at the 6th International Workshop on Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2011). Joint work with Dominic Mulligan.
6th International Workshop on Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2011) Nijmegen, Netherlands
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Talk type: Workshop talk
Two-and-a-half order lambda-calculus
Talk given on Friday 4 July in the WFLP 2008 in Italy at 10:00. I describe two-and-a-half order lambda-calculus, a two-level lambda-calculus. Normally, the lambda-calculus has a capture-avoiding substitution. The two-level lambda-calculus is actually two lambda calculi, with distinct namespaces for variable symbols of `strong’ and `weak’ variables, but sharing a common application operator. Substitution for a strong variable does not avoid capture by lambda-abstraction for a weak variable. This leads to interesting and unusual phenomena.
Workshop on Functional and Logic Programming (WFLP 2008) Siena, Italy
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Talk type: Seminar
Two-and-a-halfth order lambda-calculus: a calculus of the informal meta-level
A talk given in the DSG (Dependable Systems Group) meeting. Joint work with Dominic Mulligan.
Dependable Systems Group meeting, Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
The lambda context calculus
Given at LFMTP, part of CADE07 near Bremen on July 15 at 17:00. Joint work with Stéphane Lengrand”.
Logical Frameworks and Meta-languages: Theory and Practice (LFMTP), CADE 2007 Bremen, Germany
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Talk type: Seminar
Boxology
Given at St Andrew’s University on December 19 at 11:00. We discuss boxology and hierarchical nominal techniques. Thanks to Kevin Hammond.
University of St Andrews St Andrews, UK
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Talk type: Teaching talk
The lambda-calculus
A brief talk-ette (talk-chen, talk-ino) presented to test my presentation skills as part of the PGCAP course.
PGCAP course, Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
Incomplete objects using the NEW calculus of contexts
Given at the DREAM workshop of the mathematical reasoning group at Edinburgh University on September 1. We speculate on higher-order logic in the NEW calculus of contexts.
DREAM workshop, University of Edinburgh Edinburgh, UK
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Talk type: Course / lecture
A NEW calculus of contexts
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 12:00 on Sunday 14 May. Also see gabbay:newcc.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
The NEW calculus of contexts
Talk given at the Scottish Programming Languages Seminar on February 1. We describe the NEW calculus of contexts, a lambda-calculus which tries to understand the meta-level/object-level split. Thanks to Edwin Brady. (I gave similar talks in Turin, Italy and at PPDP05.)
Scottish Programming Languages Seminar, University of St Andrews St Andrews, UK
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Talk type: Workshop talk
The NEW calculus of contexts
Talk given at the Turin workshop on implicit computational complexity on January 19. We describe the NEW calculus of contexts, a lambda-calculus which tries to understand the meta-level/object-level split. Thanks to Simona Ronchi della Rocca. (A corrected and slightly modified version of a talk given at PPDP05.)
Workshop on Implicit Computational Complexity, University of Turin Turin, Italy
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Talk type: Conference talk
A NEW calculus of contexts
Talk given at PPDP05 in Lisbon on 10 July 2005. We discuss a lambda-calculus with a hierarchy of variables of different ‘strengths’ (corresponding roughly to increasingly ‘meta’ variables). We find it to be remarkably expressive, and suggest that we can encode records, modules, objects, dynamic binding and rebinding, and many other constructs involving non-capture-avoiding substitution.
Principles and Practice of Declarative Programming (PPDP 2005) Lisbon, Portugal
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Talk type: Seminar
A NEW calculus of contexts
Slides of a talk given in INRIA Rocquencourt Paris, on 21 February, and PPS Paris, on 22 February. Thanks to James Leifer, Russ Harmer, and the people of PPS and INRIA.
INRIA Rocquencourt Paris, France
- A NEW calculus of contexts Research seminar INRIA Rocquencourt Paris, France Slides for A NEW calculus of contexts
- A NEW calculus of contexts Research seminar PPS Paris, France Slides for A NEW calculus of contexts
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Talk type: Seminar
Incomplete lambda-terms
We use Nominal Rewriting to specify and discuss a lambda-calculus with meta-variables and explicit substitution. Talk given in LIX as part of a series of informal talks on meta-variables.
LIX Paris, France
Logic and proof theory 37 talks
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Talk type: Seminar
Arithmetising FOL
I give an accessible introduction to my paper on Arithmetisation of computation via polynomial semantics for first-order logic in the Journal of Applied Logics, Volume 12, number 6, pp 1481–1548, October 2025.
Presented by Murdoch J. Gabbay Imperial College Research Seminar
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Talk type: Seminar
SNARKs for pure logic
A presentation of zkFOL (arithmetisation of first-order logic) to Circle’s cryptography group.
Presented by Murdoch J. Gabbay Circle Research Seminar
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Talk type: Conference talk
A shallow arithmetisation of first-order validity
I discuss how to directly arithmetise (cryptographer-speak for ‘translate into polynomials, the language of cryptography’) validity over finite models. This means that you can specify a property – e.g. some desired correctness property – and via cryptographic magic generate succinct or zero-knowledge proofs that a model satisfies that correctness property, without communicating that model to the recipient of the proof. This is very useful, I would argue, for quickly proving correctness of data, behaviour, or anything else that can be finitely represented.
Presented by Murdoch J. Gabbay 41st British Colloquium for Theoretical Computer Science (BCTCS 2025) Room TL325B, Learning and Teaching Building, University of Strathclyde Hosted by University of Strathclyde Glasgow, Scotland
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Talk type: Lightning talk
Polynomial semantics for FOL
I discuss zkFOL, the arithmetisation of first-order logic validity over finite models.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2024 (SPLV24) University of Strathclyde Glasgow, Scotland
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Talk type: Conference talk
What are variables of first-order logic and the lambda-calculus?
What a nice event the Logic Colloquium was. What a treat to be surrounded by people who care so much about logic I learned a lot.
Thanks to the Logic Colloquium.
Logic Colloquium 2016 Leeds, UK
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Talk type: Seminar
What sequent quantifier rules tell us about nominal semantics for logic
A precis of parts of nominal semantics for predicate logic and semantics out of context.
Thanks to the Leeds Logic Group.
Leeds Logic Group, University of Leeds Leeds, UK
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Talk type: Seminar
The algebraisation of logic using nominal techniques
Over the past five years a number of systems, such as first-order logic and the lambda-calculus, have been algebraically axiomatised using nominal techniques. A next step is to relate these (equational) axiomatisations to order-theoretic notions such as complemented bounded lattices (a.k.a. Boolean algebras). I will talk about recent progress here, giving an order-theoretic account of first-order logic: the punchline is that it will correspond to a complemented finitely fresh-complete lattice with a substitution—wait for the talk or see my recent paper with Dowek Nominal semantics for predicate logic: algebras, substitution, quantifiers, and limits. Israel has a tradition in logical foundations. Fraenkel–the Fraenkel of Zermelo-Fraenkel and Fraenkel-Mostowski set theory–was Dean of Jerusalem University. So it was special while visiting the logic community in Israel to discover not only that they already knew the background to what I was talking about, but that also, their research had anticipated much of what I was about to say. Special thanks here to Arnon Avron.
Tel Aviv University Tel Aviv, Israel
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Talk type: Conference talk
Nominal semantics for predicate logic
Presentation at the 9th Italian Convention on Computational Logic (CILC’2012) in Rome, Italy. See also the associated paper, which is called Nominal semantics for predicate logic.
9th Italian Convention on Computational Logic (CILC 2012) Rome, Italy
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Talk type: Workshop talk
On proof-search
Talk given at the mathematically structured programming group as part of the STP (Scottish Theorem Provers) meeting. I describe upcoming research to give a semantics to meta-variables in proof-search and theorem-proving.
Scottish Theorem Provers meeting, University of Strathclyde Glasgow, UK
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Talk type: Conference talk
Metamathematics based on nominal terms: first-order logics over nominal sets
A talk given at British Colloquium for Theoretical Computer Science (BCTCS 2011) in Birmingham, UK. I discuss the application of nominal sets to designing metamathematical syntaxes.
British Colloquium for Theoretical Computer Science (BCTCS 2011) Birmingham, UK
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Talk type: Workshop talk
Metamathematics based on nominal terms and nominal logic: foundations of a nominal theorem-prover
A talk given in the Automated Reasoning Workshop in Glasgow University, UK. I describe permissive nominal terms and their application in logic and algebra.
Automated Reasoning Workshop University of Glasgow, UK
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Talk type: Workshop talk
Tattered type theories: type theories with holes
Part of the fun in the afternoon series of meetings. I describe current work with Dominic Mulligan on representing incompleteness in theorem-provers.
Thanks to Conor McBride.
Fun in the Afternoon meeting, University of Strathclyde Glasgow, UK
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Talk type: Conference talk
Permissive nominal logic and all that
Given at the 2010 British Logic Colloquium in Birmingham, UK. I describe in broad terms names and their symmetry properties.
Thanks to Eike Ritter.
British Logic Colloquium 2010 Birmingham, UK
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Talk type: Seminar
PNL: Permissive-nominal logic
Given as part of the Computer Science Seminars in Leicester, UK. I discuss permissive-nominal logic; an extension of first-order logic with names and binding.
Thanks to Alexander Kurz.
Computer Science Seminar, University of Leicester Leicester, UK
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Talk type: Seminar
Nominal terms and one-and-a-half level Curry Howard
A talk given at the Cambridge University computer lab as part of their Logic and Semantics seminar series, on Friday 26 September at 2pm. Joint work with Dominic Mulligan.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Seminar
One-and-a-halfth order Curry Howard: Incomplete Derivations
A talk given at the École Polytechnique on Tuesday 4 March at 11am. Joint work with Dominic Mulligan.
École Polytechnique / LIX Paris, France
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Talk type: Colloquium
Term sequent logics: cut your terms
Sequents, entailments, and cut-elimination — applied to term rewriting and reduction.
Presented by Michael Gabbay With Murdoch Gabbay Department of Computer Science colloquium King's College London, Department of Computer Science Hosted by King's College London London, UK
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Talk type: Workshop talk
a-logic with arrows
Workshop on Functional and Logic Programming, RDP 2007 Paris, France
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Talk type: Seminar
Resourceful truth: elementary models of resource-sensitive logics
A talk given at the LFCS logic and semantics club in Edinburgh University on June 15 at 13:00. Thanks to Bartek Klin.
Logic and Semantics Club, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Resourceful truth: elementary models of resource-sensitive logics
A talk given at Queen Mary and Westfield New College on June 4 at 15:30. Thanks to Hongseok Yang.
Queen Mary, University of London London, UK
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Talk type: Seminar
Nominal Logic
A talk given at the University of Bath Computer Science Department on November 13 at 11:15. We discuss nominal logic and the slogans ‘logic with names’ versus ‘logic with variables’. Thanks to Mark Price and Alessio Guglielmi.
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Talk type: Seminar
Logic of names and freshness (splitting the atom)
A talk given at the Dependable Systems Group weekly seminar at Heriot-Watt on October 27 at 14:15. We discuss what happens to the lambda binder in the presence of a hierarchy of variables.
Dependable Systems Group Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Conference talk
One-and-a-halfth-order-logic
Presented at PPDP06 in Venice, Italy on Tuesday 11 July 2006 at 16:30.
Principles and Practice of Declarative Programming (PPDP 2006) Venice, Italy
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Talk type: Seminar
Evolved logic: the next step
Presented at the programming languages group in Tel-Aviv University at 14:00 on Sunday 28 May. Thanks to Mooly Sagiv.
Programming Languages Group, Tel Aviv University Tel Aviv, Israel
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Talk type: Workshop talk
A concrete model of linearity and separation
Presented at the Scottish Theorem Proving Meeting at Heriot-Watt University on March 17. We describe what we think is probably a uniform sets-based semantics for linear logic and bunched implications. Thanks to Andrew Ireland. I repeated this talk as a research seminar with the Babel group at the Universidad Politecnica de Madrid on March 28 with title “A concrete semantics for truth in bunched implications and linear logic?”. Thanks to Julio Marino.
Scottish Theorem Proving Meeting, Heriot-Watt University Edinburgh, UK
- A concrete model of linearity and separation Scottish Theorem Proving Meeting Heriot-Watt University Edinburgh, UK Slides for A concrete model of linearity and separation
- A concrete semantics for truth in bunched implications and linear logic? Babel group research seminar Universidad Politécnica de Madrid Madrid, Spain Slides for A concrete semantics for truth in bunched implications and linear logic?
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Talk type: Seminar
Separation
Departmental seminar given at Heriot-Watt on March 6. We report on joint work with Joe Wells in which we consider semantics for multiplicative and additive implication. Thanks to Lilia Georgieva.
Departmental Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
Automating first-order logic
Given at the second CANS meeting on December 6. We present Nominal Algebraic Specifications, a theory of universal algebra based on Nominal Terms and thus able to algebraically specify, in a suitable sense, theories-with-binding. We show how an algebraic theory of first-order logic leads to “one-and-a-halfth-order logic”, which is above first-order logic and slightly to the side of second-order logic. We discuss proof-methods for the system and possible extensions. Thanks to Maribel Fernández.
Second CANS meeting, King's College London London, UK
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Talk type: Colloquium
One-and-a-halfth-order logic
Given at the Institute for Logic, Language, and Computation, in their colloquium on mathematical logic. We present one-and-a-halfth-order logic, which is a logic somewhere above first-order logic and slightly to the side of second-order logic. It satisfies cut-elimination, has an algebraic presentation, and allows to quantify over all predicates in an interesting and unusual way. Thanks to Benedikt Loewe.
Mathematical Logic Colloquium, Institute for Logic, Language and Computation Utrecht University, Netherlands
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Talk type: Seminar
a-logic
Talk given in the Logic Tea series at the University of Amsterdam on November 22. We discuss a-logic, a first-order logic with a predicate atom which is true of variable symbols and false of non-variable symbol terms. We axiomatise the untyped lambda-calculus by way of example and discuss what models look like, some possible future work, and the design decisions (and some of the sins) hidden in the paper. Thanks to Johan van Benthem and Olivier Roy.
Logic Tea, University of Amsterdam Amsterdam, Netherlands
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Talk type: Seminar
a-logic, the lambda-calculus, and internalising meta using names and binding
Talk given at LMU in Munich on 26 July 2005. We give a broad view of our research, mentioning a-logic and a whole load of other stuff.
LMU Munich Munich, Germany
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Talk type: Seminar
Restart: a natural rule
Talk given at South Bank University in London on 25 July 2005. We discuss Restart, a Natural Deduction rule which turns intuitionistic logic into classical logic, without affecting proof-normalisation.
London South Bank University London, UK
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Talk type: Colloquium
One quantifier for forall and exists and generic elements or ... unknowns are data
Slides of a talk given at the departmental colloquium on 24 November 2004.
Departmental Colloquium, King's College London London, UK
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Talk type: Conference talk
A Sequent Calculus for Nominal Logic
Slides of a talk at LICS04 on joint work with James Cheney, given in Turku, Finland, on 15 July 2004. See also the paper.
Logic in Computer Science (LICS 2004) Turku, Finland
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Talk type: Seminar
Fresh Logic
We discuss Fresh Logic (joint work with James Cheney) and consider its relation to FOLN a logic by Miller and Tiu, especially in the light of insights from implementation. Talk given to the PARCIFAL group at LIX on 14 May 2004.
PARCIFAL Group, LIX Paris, France
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Talk type: Workshop talk
Restart as a Computational Rule
Talk given on applying the Curry-Howard correspondence to a novel Restart Logic. Given in the Workshop on Lambda Calculus, Type Theory, and Natural Language in King’s College London, on 9 December at 12:00 on work.
Workshop on Lambda Calculus, Type Theory, and Natural Language King's College London, UK
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Talk type: Seminar
Fresh Logic: Vanilla First-Order Logic with FM?
Talk given on Fresh Logic (a first-order logic for the NEW-quantifier with proof-normalisation) on 2 June at the local group’s weekly Semantics Lunch. See also an abstract.
Semantics Lunch, University of Cambridge Cambridge, UK
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Talk type: Workshop talk
FM-HOL
Contributed talk at Workshop on Thirty Five years of Automath, Informal Proceedings, Heriot-Watt University, Edinburgh, Scotland, April 2002.
Workshop on Thirty Five Years of Automath Heriot-Watt University, Edinburgh, UK
Nominal techniques 40 talks
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Talk type: Seminar
Equivariance: the name of the true monster in your induction
My PhD was all about difficulties encountered combining inductively defined syntax with variable binding. In this talk I make two arguments: first, that names are best considered a primitive datatype with specific mathematical properties; and second, that the specific property is fundamentally not the fancy stuff that most of my PhD was about, but instead it is the apparently bland assertion of equivariance, namely, the property that a name-permutation universally applied, preserves validity. Once you have equivariance, I argue, the rest of the theory drops out. Conversely, if you do not take equivariance as primitive, your treatment of names may become painful. In short: equivariance is it. For more on these ideas, see the associated paper, the Nominal Haskell package, and the Nominal Agda package.
Presented by Murdoch J. Gabbay LAIV seminar
- Slides for Equivariance: the name of the true monster in your induction
- Related paper for Equivariance: the name of the true monster in your induction
- Nominal Haskell package for Equivariance: the name of the true monster in your induction
- Nominal Agda package for Equivariance: the name of the true monster in your induction
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Talk type: Conference talk
Nominal techniques as an Agda library
Nominal techniques … in Agda! Proof-of-concept but a promising start. If you like this and want to develop it further or transfer the ideas to (say) Lean, then please get in touch.
Presented by Orestis Melkonian With Murdoch J. Gabbay 29th International Conference on Types for Proofs and Programs (TYPES 2023) ETSI Informàtica, Universitat Politècnica de València Hosted by Universitat Politècnica de València Valencia, Spain
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Talk type: Seminar
Nominal programming for the working programmer
What it takes to implement a nominal package in a programming language, drawing on the Nominal Haskell package.
Presented by Murdoch J. Gabbay Scottish Programming Languages Seminar (SPLS) Hosted by University of Strathclyde Online
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Talk type: Seminar
The Design of the Nominal Datatypes Package in Haskell
Introducing: a Nominal monad
Thanks to the Huawei-Edinburgh Joint Lab. Seminar details and video here.
Huawei–Edinburgh Joint Lab Edinburgh, UK
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Talk type: Course / lecture
Nominal Datatypes in Haskell
We demo my Nominal Datatypes Haskell package
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2020 (SPLV20) Laboratory for Foundations of Computer Science, University of Edinburgh Hosted by University of Edinburgh Online
- Nominal Datatypes in Haskell — Lecture 1 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 1
- Nominal Datatypes in Haskell — Lecture 2 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 2
- Nominal Datatypes in Haskell — Lecture 3 SPLV 2020 — Nominal datatypes Laboratory for Foundations of Computer Science, University of Edinburgh Online Recording for Nominal Datatypes in Haskell — Lecture 3
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Talk type: Course / lecture
Programming with nominal techniques
Outline of a nominal datatypes package.
Thanks to the summer school organisers. See the summer school webpage.
International Summer School on Metaprogramming 2019 Schloss Dagstuhl, Germany
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Talk type: Conference talk
Nominal techniques: past and future
An observation of nominal techniques is that names are a datatype
Thanks to the ICALP organisers. See the ICALP programme.
ICALP 2019 Patras, Greece
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Talk type: Seminar
Nominal: the big picture
Thanks to the Birmingham Theory Seminars.
Birmingham Theory Seminar University of Birmingham, UK
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Talk type: Seminar
A (partial) survey of nominal techniques
Just as the title says!
Thanks to Neil Ghani and his lively and interesting group for the very enjoyable afternoon.
University of Strathclyde Glasgow, UK
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Talk type: Seminar
An informal survey of nominal techniques
I sketch recent activity in nominal techniques and speculate on how nominal techniques could be useful in verification. The talk was aimed especially those not familiar with nominal techniques. I gave the talk without slides.
IMDEA Software Institute Madrid, Spain
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Talk type: Seminar
The application of symmetry to computing and meta-mathematics
Symmetry has long been recognised as important in mathematics and physics, but its importance has been less appreciated in theories of logic and computation. In this (mathematically oriented, but non-technical) talk I describe what symmetry means in logic and computation, examine the benefits we can obtain by considering logic and computation as inherently “symmetric” mathematical entities, and relate this back to traditional foundations based on sets and functions. Special thanks here to Stuart Margolis and the remarkable Plotkin family.
Bar-Ilan University Tel Aviv, Israel
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Talk type: Seminar
An overview of nominal techniques, semantics, and logic
Given at the Artificial Intelligence Research Institute (IIIA) in Barcelona, Spain. Does what it says on the tin. I gave the talk on the whiteboard; there are no slides.
Artificial Intelligence Research Institute (IIIA) Barcelona, Spain
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Talk type: Seminar
Nominal techniques: a success story of the continued relevance of mathematical foundations to computer science
A talk given at Glasgow University maths department. I discuss nominal techniques from a foundational point of view.
Mathematics Seminar, University of Glasgow Glasgow, UK
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Talk type: Seminar
A gentle introduction to the shape of nominal techniques
The same beginner-friendly talk I gave in February. I gave the talk on the whiteboard; there are no slides.
University of Strathclyde Glasgow, UK
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Talk type: Seminar
A gentle introduction to the shape of nominal techniques
A beginner-friendly talk given at the University of Glasgow, UK on nominal techniques. The abstract and announcement are here. I gave the talk on the whiteboard; there are no slides.
University of Glasgow Glasgow, UK
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Talk type: Seminar
Nominal techniques, 2006-2011
Given at Tel-Aviv university. I broadly survey of the development of nominal terms in logic since 2006 when I last visited the department, discussing permissive-nominal algebra and permissive-nominal logic. I gave the talk on the whiteboard; there are no slides.
Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Nominal techniques: logique, algèbre, calcul
A talk given at the 16th meeting on Logic, Algebra, and Calculus at PPS in Paris, France. I talk about applications of nominal rewriting, algebra, and permissive-nominal logic to specifying and reasoning about logic, algebra, and calculus.
Thanks to Delia Kesner and DIGITEO.
16th meeting on Logic, Algebra, and Calculus, PPS Paris, France
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Talk type: Seminar
Nominal sets and the NEW-quantifier
A talk given at the Typical Seminar at the École Polytechnique in Paris, France. I describe recent work on a representation theorem for the NEW-quantifier.
Thanks to Gilles Dowek and DIGITEO.
Typical Seminar, LIX, École Polytechnique Paris, France
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Talk type: Workshop talk
Permissive nominal terms
A talk given at the Scottish Theorem Proving meeting in Scotland, UK. I discuss permissive nominal terms. I used the whiteboard; there are no slides.
Thanks Grant Passmore.
Scottish Theorem Proving meeting, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
Names: I denote, therefore I am
A talk given at the SICSA Complex Systems Engineering Joint THREADSS/SEMANTICS Meeting in Scotland, UK. I discuss in general (philosophical?) terms some of the ideas and motivations underlying my research.
Thanks to Phil Trinder.
SICSA Complex Systems Engineering Joint THREADSS/SEMANTICS Meeting Heriot-Watt University, Edinburgh, UK
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Talk type: Seminar
Permissive nominal terms
A 30 minute ‘theory lunch’ seminar given at IMDEA in Madrid, Spain. I discuss permissive nominal terms. I used the whiteboard; there are no slides.
Thanks to Alejandro Sanchez.
Theory Lunch, IMDEA Software Institute Madrid, Spain
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Talk type: Course / lecture
Nominal techniques: a theory of names and binding
A series of three 90 minute talks on nominal techniques given at the Universidad Complutense in Madrid, Spain. I used the whiteboard; there are no slides.
Thanks to Yolanda Ortega Mallen.
Universidad Complutense de Madrid Madrid, Spain
- Nominal techniques: a theory of names and binding 6–22 October 2009 (three 90-minute lectures; individual dates not recorded) Nominal-techniques lecture series Universidad Complutense de Madrid Madrid, Spain
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Talk type: Seminar
Nominal techniques: present, and future prospects
Given in the Prose seminar series at TU/e in Eindhoven on November 14 at 15:30. I talk about where I think things should be going.
Prose seminar series, Eindhoven University of Technology Eindhoven, Netherlands
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Talk type: Workshop talk
Names, computations, logics: the hole story.
Given at the Mathematical Theories of Abstraction, Substitution and Naming in Computer Science in ICMS, Edinburgh on May 27 at 12:05. Thanks to the organisers Fernández, Pitts, and Stark. I describe my current take on ‘nominal techniques’.
Mathematical Theories of Abstraction, Substitution and Naming in Computer Science ICMS, Edinburgh, UK
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Talk type: Seminar
Denotations for functional programs with names
I consider denotations for functional programming in substitution sets, an algebraic semantics for substitution.
Presented by Murdoch J. Gabbay DSG Digest, Heriot-Watt University Edinburgh, Scotland
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Talk type: Seminar
Fraenkel-Mostowski atoms model variables as well as names
A talk given at SPLS in Edinburgh on March 8 at 14:30. In this non-technical talk, we discuss a recent paper in which I exhibit a capture-avoiding substitution action on the Fraenkel-Mostowski sets universe. This suggests a new view of variables as denotational objects (names with a substitution action). Thanks to Kenneth MacKenzie.
Scottish Programming Languages Seminar, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Names and variables
A talk given at City University on February 27 at 12:00. We discuss our research intentions to investigate denotations in which names are first-class denotational citizens; why one might want to do this, and how we are doing it. Thanks to Artur Garcez.
City University London, UK
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Talk type: Seminar
Fraenkel-Mostowski atoms model variables as well as names
A talk given at Heriot-Watt University on February 20 at 14:00. We discuss a recent paper in which I exhibit a substitution action on the Fraenkel-Mostowski sets universe. Thanks to Pierluigi Frisco.
Heriot-Watt University Edinburgh, UK
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Talk type: Workshop talk
Nominal techniques: evolving the atom
A talk given at the Scottish Theorem Proving meeting in the School of computer science. We give a position talk on nominal techniques considered useful as a means to generalise compositionality. Thanks to Phil Trinder and Rob Rothenberg.
Scottish Theorem Proving meeting, University of St Andrews St Andrews, UK
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Talk type: Seminar
Nominal: an overview
Presented at the Departmental Seminar at the Universidad Politecnica de Madrid on March 31. We give an overview of (one large part of) our research, focussing on ‘nominal’. Thanks to Manuel Hermenegildo, Juan-Jose Moreno, and Jim Lipton for the invitation to visit.
Departmental Seminar, Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Seminar
Nominal techniques and the meta-level
Talk given at Venice University on January 24. In reaction to the question “So what is it that you do?”, I give an initially non-technical description of the motivations (as I see them) for that part of my work to date which has to do with nominal techniques. Thanks to Annalisa Bossi.
University of Venice Venice, Italy
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Talk type: Seminar
The Fraenkel-Mostowski (FM) model of abstraction applied to name-generation
Given at the ULTRA group seminar at Heriot-Watt on January 9. We discuss the FM model of abstraction, and briefly suggest its usefulness for both inductive and coinductive types. Thanks to Lilia Georgieva.
ULTRA Group Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Seminar
Nominal Logic and FM techniques: reasoning about binding without pain
Given at the Foundations Group seminar of the Brouwer Institute in Radboud University in Nijmegen on November 29. We give a relatively technical sketch of FM techniques, Nominal Logic, and their application to reasoning on syntax-with-binding. Thanks to Henk Barendregt and Bas Spitters. I gave the same talk at the Logic and Semantics Group Joint Theory seminar (‘Joint’ between Queen Mary and Westfield College and Imperial College) at Queen Mary and Westfield College on December 7 under the name “Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction”. Thanks to Peter O’Hearn. One member of the audience in Nijmegen then kindly invited me to give the talk again at the Theoretical Computer Science department in the Vrije Universiteit Amsterdam on December 12 (I used the original title). Thanks to Jan-Willem Klop and Femke van Raamsdonk.
Foundations Group Seminar, Brouwer Institute Radboud University, Nijmegen, Netherlands
- Nominal Logic and FM techniques: reasoning about binding without pain Foundations Group seminar, Brouwer Institute Radboud University Nijmegen, Netherlands Slides for Nominal Logic and FM techniques: reasoning about binding without pain
- Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction Logic and Semantics Group Joint Theory seminar Queen Mary and Westfield College London, UK Slides for Fraenkel-Mostowski Techniques, a NEW model of freshness and abstraction
- Nominal Logic and FM techniques: reasoning about binding without pain Theoretical Computer Science seminar Vrije Universiteit Amsterdam Amsterdam, Netherlands Slides for Nominal Logic and FM techniques: reasoning about binding without pain
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Talk type: Workshop talk
Nominal terms with a hierarchy of variables
Bleeding-edge talk given in Eindhoven in a group workshop on 31 October 2005. We give a speculative account of an algebraic version of the NEW calculus of contexts.
Group workshop, Eindhoven University of Technology Eindhoven, Netherlands
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Talk type: Seminar
Nominal Terms, Existential Variables, and Mathematics
Talk given at PPS on 4 May 2004.
PPS Paris, France
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Talk type: Research talk
Extensions of Nominal Terms
We discuss extensions of Nominal Terms (see the paper on Nominal Unification). Lots of blue sky, but pretty.
Not recorded in the legacy archive
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Talk type: Seminar
FM for Names and Binding
Talk given on 11/12/2002 at the Japan Advanced Institute of Science and Technology, Ishikawa, on 14/12/2002 at MLG 2002 in Kinosaki, and on 16/12/2002 at ALGI in Kyoto. Thanks to René Vestergaard.
Japan Advanced Institute of Science and Technology Ishikawa, Japan
- FM for Names and Binding Research seminar Japan Advanced Institute of Science and Technology Ishikawa, Japan Slides for FM for Names and Binding
- FM for Names and Binding MLG 2002 Kinosaki, Japan Slides for FM for Names and Binding
- FM for Names and Binding ALGI Kyoto, Japan Slides for FM for Names and Binding
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at the Max-Planck Institute, Saarbrücken, Germany.
Max Planck Institute Saarbrücken, Germany
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at INRIA Rocquencourt France 11/10/02 and Université Paris VII France 14/10/02. Thanks to Gilles Dowek and Pierre-Louis Curien.
INRIA Rocquencourt and Université Paris VII Paris, France
- FM Techniques for Syntax-with-Binding Research seminar INRIA Rocquencourt Paris, France Slides for FM Techniques for Syntax-with-Binding
- FM Techniques for Syntax-with-Binding Research seminar Université Paris VII Paris, France Slides for FM Techniques for Syntax-with-Binding
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Talk type: Seminar
FM Techniques for Syntax-with-Binding
Talk given at IBM Research, Haifa, Israel 2/10/02. Thanks to Cindy Eisner.
IBM Research Haifa, Israel
Process calculi 5 talks
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Talk type: Seminar
FM binding for pi-calculus transitions
Talk given on May 30 at Fundacao da Faculdade de Ciencias e Tecnologia / University of Lisbon. Thanks to Luis Caires and his department. See also an abstract.
University of Lisbon Lisbon, Portugal
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Talk type: Seminar
FM and the pi-calculus
I investigate what happens if we consider an FM-like category of objects with a finite name-for-name renaming action (possibly non-injective maps between names) rather than a finite name-for-name permutation action as is the case for FM sets. I obtain a category similar to Schan, but which supports a natural interpretation of weak HOAS. Talk given on March 20 at the ENS Lyon, France. Thanks to Daniel Hirschkoff. See also an abstract.
École Normale Supérieure de Lyon Lyon, France
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Talk type: Seminar
Restriction, binding, and three presentations of the pi-calculus
I’ve been using FreshML to program a pet interest of mine; transition systems and bisimulation checkers for the pi-calculus (see [gabbay:thempc]). I’ll show excerpts from the programs I’ve written and talk about what writing them has taught me about another interest of mine; finding a semantics for restriction as distinct from binding. The sourcecode discussed is the pi-ltsb series of programs above, see [pi-ltsb] above.
Semantics Lunch, University of Cambridge Cambridge, UK
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Talk type: Seminar
Models of pi-calculus behaviour using FM techniques
Talk given in Pisa on 2/12/2002. Thanks to Ugo Montanari.
University of Pisa Pisa, Italy
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Talk type: Seminar
Theory and Models of the pi-calculus using Fraenkel-Mostowski Generalised
Talk given in Cambridge on 11/11/2002 at the local group’s “Semantics Lunch”.
Semantics Lunch, University of Cambridge Cambridge, UK
Programming languages 2 talks
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Talk type: Workshop talk
The Cairo Test Suite Workshop
An online presentation of my Cairo property-based and unit test suite (also on GitHub).
Thanks to Matchbox DAO for the kind invitation to speak.
Matchbox DAO Online
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Talk type: Seminar
FreshML: programming with binders made easy
Talk given on May 28 at Fundacao da Faculdade de Ciencias e Tecnologia / University of Lisbon. Thanks to Luis Caires and his department. See also an abstract.
University of Lisbon Lisbon, Portugal
Rewriting and unification 16 talks
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Talk type: Workshop talk
Closed nominal rewriting and efficiently computable nominal algebra equality
A talk given at the 5th International Workshop on Logical Frameworks and Meta-languages (LFMTP 2010) in Edinburgh, UK. I discuss the connection between nominal rewriting and nominal algebra. Joint work with Maribel Fernández.
5th International Workshop on Logical Frameworks and Meta-languages (LFMTP 2010) Edinburgh, UK
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Talk type: Conference talk
Permissive nominal terms and their unification
A talk given at CILC09, the 24th Italian Conference on Computational Logic in Ferrara, Italy. See also the associated paper. I discuss permissive-nominal terms, which are similar to nominal terms but with properties more like those of first-order syntax. In particular, permissive-nominal terms lack freshness contexts, have inherent notions of alpha-equivalence and ‘free atoms of’, and the notion of solution of a unification or matching problem is based just on substitution.
24th Italian Conference on Computational Logic (CILC 2009) Ferrara, Italy
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Talk type: Workshop talk
Hierarchical nominal rewriting
Kindly presented on my behalf by James Cheney at LFMTP06 in Seattle, USA on Wednesday 16 August 2006 at 10:30.
Presented by James Cheney Logical Frameworks and Meta-languages: Theory and Practice (LFMTP 2006) Seattle, USA
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Talk type: Course / lecture
On substitution
Presented as part of a course in Tel-Aviv University in 210 Schreiber at 10:30 on Thursday 11 May. Also see gabbay:capasn and gabbay:subaan.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Functional programming and unification with meta-variables
A research seminar given with the Babel group at the Universidad Politecnica de Madrid on March 31. We discuss our recent axiomatisation of substitution, and the NEW calculus of contexts, suggesting connections with logic programming and describing what we think might naturally be done next in this area. Thanks to Julio Marino.
Babel Group Seminar, Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Colloquium
Is substitution an abstract notion?
Departmental colloquium of March 13. We describe an axiomatisation of substitution in ‘nominal style’ and discuss its set-based semantics. Thanks to Iman Poernomo.
Departmental Colloquium, King's College London London, UK
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Talk type: Course / lecture
A NEW semantics of substitution
Report on joint research carried out with two students, Samuel Rota Bulo and Andrea Marin, in the framework of a course taught in the University of Venice. Presented by the students on March 1. Thanks to their tutor Simonetta Balsamo.
Presented by Samuel Rota Bulo and Andrea Marin University of Venice Venice, Italy
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Talk type: Colloquium
Nominal Rewriting
Talk given at the computational logic colloquium at Innsbruck University on January 23. We describe nominal rewriting as a meta-level framework for specifying rewriting, which accurately reflects the instantiation behaviour of meta-level variables with respect to object-level binding. We then expatiate way over the time limit on the general context of our research. Thanks to Aart Middeldorp for inviting me and to Georg Moser for making it happen.
Computational Logic Colloquium, University of Innsbruck Innsbruck, Austria
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Talk type: Seminar
Nominal Rewriting. Or ... talking about functions, without functions
Talk given at Tel-Aviv University on 6 October 2005. We give a non-technical account of Nominal Rewriting. Thanks to Arnon Avron and Nachum Dershowitz.
Tel Aviv University Tel Aviv, Israel
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Talk type: Seminar
Nominal Rewriting
Talk given at Leicester University on 9 August 2005. We discuss Nominal Rewriting and suggest why it and the methods underlying it are of interest.
University of Leicester Leicester, UK
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Talk type: Conference talk
Extended Nominal Rewriting: two models of binding
Talk given at PPDP05 in Lisbon on 10 July 2005. We discuss two notions of binding which exist side-by-side in Extended Nominal Rewriting; one modelling lambda-calculus-like abstraction, the other modelling pi-calculus-like restriction. Some typos fixed with respect to the similar talk in Paris, and only 30 minutes long.
Principles and Practice of Declarative Programming (PPDP 2005) Lisbon, Portugal
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Talk type: Seminar
Extended Nominal Rewriting: two models of binding
Talk given at LIX, Ecole Polytechnique, France, on 24 May. Thanks to Dale Miller. We discuss two notions of binding which exist side-by-side in Extended Nominal Rewriting; one modelling lambda-calculus-like abstraction, the other modelling pi-calculus-like restriction.
LIX, École Polytechnique Paris, France
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Talk type: Seminar
Aspects of Nominal Rewriting
Slides of a talk given at the CWI in Amsterdam on 2 March, and TUE in Eindhoven on 3 March. Thanks to Anton Wijs and MohammadReza Mousavi.
CWI Amsterdam, Netherlands
- Aspects of Nominal Rewriting Research seminar Centrum Wiskunde & Informatica (CWI) Amsterdam, Netherlands Slides for Aspects of Nominal Rewriting
- Aspects of Nominal Rewriting Research seminar Eindhoven University of Technology Eindhoven, Netherlands Slides for Aspects of Nominal Rewriting
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Talk type: Seminar
Nominal Unification
Slides of a talk given in Imperial College, London, on 10 September 2004. Thanks to Stephen Muggleton.
Imperial College London London, UK
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Talk type: Conference talk
Nominal Rewriting Systems
Slides of a talk at PPDP04 on joint work with Maribel Fernández and Ian Mackie, given in Verona, Italy, on 26 August 2004. See also the paper.
Principles and Practice of Declarative Programming (PPDP 2004) Verona, Italy
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Talk type: Seminar
Nominal Rewriting
We present Nominal Rewriting, an essentially first-order rewriting system with inbuilt support for alpha-equivalence. See also “Incomplete lambda-terms” below. Talk given in LRI in their seminar series (27/2/2004, 10:30am).
LRI Seminar France
Semantics and duality 13 talks
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Talk type: Seminar
Topological nominal semantics
Stone duality in a nominal universe.
Thanks to Dan Marsden and Luke Ong. I outline two papers on algebraic and topological semantics for the lambda-calculus and first-order logic. See the announcement and abstract.
OASIS seminar, Department of Computer Science, University of Oxford Oxford, UK
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Talk type: Workshop talk
Nominal duality theory for new, forall, and lambda
I was privileged to be invited to the rather wonderful Dagstuhl seminar 13422 on Nominal Computation Theory in Dagstuhl, Germany.
Thanks to the organisers for this great event. Shall we do another one?
Dagstuhl Seminar 13422: Nominal Computation Theory Schloss Dagstuhl, Germany
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Talk type: Seminar
An overview of nominal algebra, lattice, representation and dualities for computer science foundations
Presentation at the Logic and Semantics Seminar in the Computer Lab, Cambridge. I discuss how nominal algebra and powersets naturally model first-order logic with equality and the lambda-calculus. We can base theories of lattices, algebra, and topology on these observations, extending the familiar style of Boolean and Heyting algebras and their representations and topologies.
Thanks to Jonathan Hayman.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Conference talk
Semantics of FOL and lambda-calculus using nominal techniques
Presentation at the Samson@60 conference in honour of Samson Abramsky. Getting to this event was unusually challenging and so I have an unusually long list of kind, generous, and efficient people to thank for my being able to attend: thanks to Luke Ong, Prakash Pananganden, Dave Corne, and Destiny Chen.
Samson@60 conference University of Oxford, UK
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Talk type: Workshop talk
Stone duality for first-order logic, a nominal approach
Presentation at the Workshop on Duality Theory in Algebra, Logic and Computer Science in the Mathematical Institute, Oxford. See also the associated paper Stone duality for first-order logic: a nominal approach to logic and topology. This was a particularly nice do, and a lovely chance for me to rub shoulders with a new community. One particular pleasure of the workshop was meeting a Polish academic who as a young man had studied with Mostowski. He told me some stories about the great man, and said how nice it was to see this mathematics being used.
Workshop on Duality Theory in Algebra, Logic and Computer Science Mathematical Institute, Oxford, UK
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Talk type: Seminar
Three semantics for logic out of context
It is common to reason about semantics in a context. This context may be a valuation (assigning denotations to variables) or a typing context (assigning types to variables), or a memory state (enumerating known memory), and so on. I will outline a nominal approach to semantics which subsumes context into a more abstract nominal notion of support. Using this we can give three semantics to (first-order) logic: in lattices, sets, and algebras. The full details are technical and in this talk I will try to bring out the non-technical “meaning of it all”, with an emphasis (necessarily speculative, but realistic) on practical implications to logic, specification, and programming.
Universidad Politécnica de Madrid Madrid, Spain
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Talk type: Workshop talk
Stone duality for first-order logic, a nominal approach
Given at the Howard Barringer workshop. Not a repeat of my talks at Paris or Edinburgh (see below), because I only had 20 minutes. Few technical details, much more discussion of the “conceptual stack” of nominal techniques and how I plan to develop the field. A very nice and interesting event with many varied speakers. The Festschrift is where the accompanying paper appears.
Howard Barringer workshop, University of Manchester Manchester, UK
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Talk type: Seminar
Stone duality for first-order logic, a nominal approach
A talk given at the work group on Types and Realisability / Théorie des types et réalisabilité”. A repeat of my talk of the previous week in Edinburgh, but to a different audience (and getting a different set of very constructive comments). Interesting to present the same work to two different audiences and see the responses.
Types and Realisability working group, INRIA Paris, France
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Talk type: Seminar
Stone duality for first-order logic, a nominal approach
Part of the LFCS seminar series in Edinburgh University. A brand-new duality result between appropriate nominal notions of topological space, and the nominal algebra axiomatisation of first-order logic from One-and-a-halfth order logic and Permissive-nominal logic.
LFCS seminar series, University of Edinburgh Edinburgh, UK
- Slides for Stone duality for first-order logic, a nominal approach
- Related paper for Stone duality for first-order logic, a nominal approach
- One-and-a-halfth-order logic for Stone duality for first-order logic, a nominal approach
- Permissive Nominal Logic for Stone duality for first-order logic, a nominal approach
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Talk type: Workshop talk
Kripke-style models in which logic and computation have equal standing
I discuss Kripke models of the untyped lambda-calculus at the lovely and exciting meeting in honour of Dana Scott.
Scott in Scotland meeting, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
Semantic Nominal Terms
A talk given at the 2nd International Workshop on Theory and Applications of Abstraction, Substitution and Naming in York, UK (a satellite event of ETAPS 2009). See also the associated paper. I discuss a semantics for nominal terms unknowns as ‘infinite streams of distinct atoms’. This reconciles atoms-abstraction with nominal terms. I also describe a form of substitution that is both capturing (as in nominal terms) and capture-avoiding (as in first-order syntax with binding) — at the same time.
2nd International Workshop on Theory and Applications of Abstraction, Substitution and Naming University of York, UK
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Talk type: Colloquium
Semantics of names and binding
A King’s College London departmental colloquium given on January 14 at noon. I gave this talk the old-fashioned way on the blackboard; hence, no pdf slides online. It was good fun.
Thanks to Iman Poernomo.
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Talk type: Seminar
FM for Higher-Order Abstract Syntax
I investigate what happens if we consider an FM-like category of objects with a finite name-for-name renaming action (possibly non-injective maps between names) rather than a finite name-for-name permutation action as is the case for FM sets. I obtain a category similar to Schan, but which supports a natural interpretation of weak HOAS. Talk given on March 13 in Manchester University, thanks to Peter Aczel. See also an abstract.
University of Manchester Manchester, UK
Semitopology and semiframes 9 talks
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Talk type: Workshop talk
Declarative Bracha Broadcast using 3-valued logic
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour. The result is simple and highly effective.
Presented by Murdoch J. Gabbay Epistemic logic and topological perspectives on distributed computing (CELT 2026) IRIF Paris, France
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Talk type: Seminar
Declarative distributed protocols using three-valued modal logic and semitopologies
Distributed protocols are typically described using imperative pseudocode. But what if we could specify them declaratively, as logical theories? We use a three-valued logic, where the third truth-value ‘b’ represents broken/Byzantine behaviour, and semitopologies provide a language for describing quorums. The result is to exhibit protocols as simple and compact axiomatic theories.
Presented by Murdoch J. Gabbay Paris Automata and Concurrency Theory Seminar Paris, France
- Slides for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Bracha Broadcast for Declarative distributed protocols using three-valued modal logic and semitopologies
- Declarative Paxos for Declarative distributed protocols using three-valued modal logic and semitopologies
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Talk type: Workshop talk
Semitopologies and semiframes for distributed heterogeneous consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). Semitopologies naturally generate a three-valued logic (by considering continuous functions out of them), and they can be dualised to semiframes, which are complete join-semilattices with a compatibility relation. We study these, along with various well-behavedness conditions on points in a semitopology.
Presented by Murdoch J. Gabbay First Workshop on Heterogeneous Trust in Distributed Systems (HTDS 2023) Convocation Room 113, Friend Center Princeton, New Jersey, USA
- Slides for Semitopologies and semiframes for distributed heterogeneous consensus
- Recording of Semitopologies and semiframes for distributed heterogeneous consensus
- Related paper for Semitopologies and semiframes for distributed heterogeneous consensus
- Semiframes paper for Semitopologies and semiframes for distributed heterogeneous consensus
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Talk type: Workshop talk
Semitopology and (paraconsistent) logic
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open (this arises from modelling distributed systems). It turns out that semitopologies are most easily classified using a three-valued paraconsistent logic; put another way, when you try to develop a logic over semitopologies, with semantics as continuous functions, a third truth-value very naturally appears.
Presented by Murdoch J. Gabbay Dagstuhl Seminar 23272: Epistemic and Topological Reasoning in Distributed Systems Schloss Dagstuhl Dagstuhl, Germany
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Talk type: Seminar
Semitopology: a new topological model of heterogeneous consensus
A quorum in a distributed system is a set of participants with sufficient power to take collective action, independently from the rest of the system. Quorums are closed under arbitrary union, but not under arbitrary intersection. It turns out that if you treat this as something that is nearly (but not quite) a point-set topology, and then replay topological notions like open set, closed set, dense set, and continuous function, interpreted as quorums, then you obtain constructs with real meaning in the theory of distributed systems (e.g. a dense set corresponds to a blocking set). It turns out further that if you take notions from distributed systems and view them topologically, then further rich structure is revealed.
Presented by Murdoch J. Gabbay Scottish Programming Languages Seminar (SPLS) Coats Building, University of the West of Scotland Paisley, Scotland
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Talk type: Seminar
The semitopology of heterogeneous consensus
A mathematical notion of consensus in a heterogeneous permissionless environment. ‘Permissionless’ means that participants are free to join or leave the network without permission from a central authority; ‘heterogeneous’ means that participants may have different goals and beliefs.
Thanks to the Online Social Choice and Welfare seminar series (permalink). Video online here.
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Talk type: Conference talk
The (semi)topology of distributed consensus
A semitopology is like a topology, but without the restriction that an intersection of open sets needs to be open. This generalises the notion of quorum from distributed systems. It soon becomes clear that the most interesting semitopologies are those that satisfy an intersection-of-open-sets property which is the precise negation of Hausdorff separation.
Presented by Murdoch J. Gabbay 11th International Conference on Geometric and Topological Methods in Computer Science (GETCO 2022) EPITA School of Engineering and Computer Science Le Kremlin-Bicêtre, France / hybrid
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Talk type: Tech talk
A semi-topological view of real-world consensus
Consensus = Topology
Thanks to the Galois talk organisers. Video online here.
Galois Tech Talk series USA / online
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Talk type: Workshop talk
A semi-topological view of real-world consensus
Consensus = Topology Thanks to the FRIDA organisers.
7th Workshop on Formal Reasoning in Distributed Algorithms (FRIDA 2020) Online
Set theory and foundations 20 talks
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Talk type: Seminar
An overview of my claimed proof of consistency of Quine's New Foundations (ConNF)
As the title suggests. The claimed proof is here.
Thanks to the LFCS seminar series. Video online here.
LFCS seminar series, University of Edinburgh Edinburgh, UK
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Talk type: Workshop talk
Equivariant ZFA and the foundations of nominal techniques
On the foundations of nominal techniques
Thanks to the summer school organisers.
XIII Summer Workshop in Mathematics Hosted by Universidade de Brasília Online
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Talk type: Seminar
The language of stratified sets, Quine's NF, rewriting, and higher-order logic
A somewhat meandering tour of some elementary and not-so-elementary results
Thanks to Jessica Ryan. See the announcement and abstract.
FATA seminar, Department of Computer Science, University of Glasgow Glasgow, UK
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Talk type: Workshop talk
Equivariant ZFA with Choice (EZFAC)
I argue for EZFAC as a foundation for nominal techniques.
Thanks to the organisers for this lovely event. Here are the ARW conference proceedings, my poster, and my slides. A longer paper is here.
Automated Reasoning Workshop 2018 Computer Laboratory, Cambridge, UK
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Just a lovely, friendly event.
Thanks to the SPLS, and to the Departmental Seminar series of Strathclyde.
Scottish Programming Languages Seminar, University of Strathclyde Glasgow, UK
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Talk type: Conference talk
Consistency of Quine's NF using nominal techniques
My first time in Brussels. It seemed an extremly nice place and was surprisingly laid-back. The Highlights Conference was lovely and I will go next year (in London) if I can. In spite of the similar title, this talk was not a pure repeat of my talks at the Logic Colloquium in August (see below). Thanks to the Highlights Conference 2016, and to Slawomir Lasota for inviting me to speak in his session.
Highlights Conference 2016 Brussels, Belgium
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Talk type: Conference talk
Consistency of Quine's NF using nominal techniques
This was the second of two connected presentations; see next talk. Thanks to the Logic Colloquium, and to Nicola Gambino for being a gentleman.
Logic Colloquium 2016 Leeds, UK
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Talk type: Seminar
Nominal techniques and consistency of Quine's NF
Thanks to the LFCS Seminars series.
LFCS seminar series, University of Edinburgh Edinburgh, UK
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Talk type: Seminar
Nominal foundations of mathematics
Thanks to the Heriot-Watt Computer Science Seminar Series.
Computer Science Seminar, Heriot-Watt University Edinburgh, UK
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Talk type: Colloquium
Consistency of Quine's NF using nominal techniques
Thanks to the King’s College London Informatics Research Colloquium Series.
Informatics Research Colloquium, King's College London London, UK
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Thanks to the Manchester Logic Seminar series.
Manchester Logic Seminar, University of Manchester Manchester, UK
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
Thanks to the Logic and Semantics seminar series.
Logic and Semantics Seminar, University of Cambridge Cambridge, UK
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Talk type: Seminar
Consistency of Quine’s NF using nominal techniques
Presented by Murdoch J. Gabbay Midlands Logic Seminar
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Talk type: Seminar
Consistency of Quine's NF using nominal techniques
See the announcement.
Thanks to the Leeds Logic Group.
Leeds Logic Group, University of Leeds Leeds, UK
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Talk type: Seminar
You are not a number: how nameless foundations took over the world and made us think we have only numbers, and no names.
Seminar at the School of Natural and Computing Sciences at the University of Aberdeen. I describe the nature of and motivation for nominal research.
School of Natural and Computing Sciences, University of Aberdeen Aberdeen, UK
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Talk type: Seminar
The nature of sets in computer science
A talk given at King’s College as part of their Advanced Programming Techniques seminar series, on Monday 29 September at 12:00. I show how from a belief in the empty set and in ordinals we can construct a mathematical universe large enough that mathematical objects (in particular, numbers, rationals, and reals) can be interpreted in it. In response to questions from the audience I discuss the difference between structure (which exists in the model) and meaning (which exists in our own minds). It was a pretty good audience!
Advanced Programming Techniques Seminar, King's College London London, UK
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Talk type: Conference talk
Substitution for Fraenkel-Mostowski Foundations
Talk given on Friday 4 April in the AISB conference in Aberdeen at 17:30. I describe a substitution action defined for the atoms on Fraenkel-Mostowski sets (the general set theory version of nominal sets). In the original Gabbay-Pitts ‘nominal’ approach to modelling syntax, the atoms in Fraenkel-Mostowski set theory were used to model variable symbols. Since they have a substitution action, it may be that they can be used to model variables, as well. Joint work with Michael Gabbay. The associated paper is here.
AISB Convention University of Aberdeen, UK
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Talk type: Workshop talk
Arbitrary objects in mathematics and semantics
Talk delivered by Michael Gabbay in the Paris Arché workshop on abstract objects in semantics and the philosophy of mathematics on Thursday February 28. This talk was delivered by my co-author Michael Gabbay; a photo of Michael delivering the talk is online here. We connect relatively recent developments in Fraenkel-Mostowski sets and nominal style semantics, with existing philosophical thinking on arbitrary-ness and unknown entities in language, mathematics, and semantics. Joint work with Michael Gabbay. The associated paper is here.
Presented by Michael Gabbay Arché workshop on abstract objects in semantics and the philosophy of mathematics Paris, France
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Talk type: Workshop talk
Substitution in Fraenkel-Mostowski sets
Given at the Centre for Mathematical Studies as part of the Cameleon meeting on December 2 at 16:00. We show how the model of names given by atoms in Fraenkel-Mostowski sets can be extended with a substitution action to a model of variables. Thanks to Thomas Forster.
Cameleon meeting, Centre for Mathematical Sciences, University of Cambridge Cambridge, UK
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Talk type: Course / lecture
Fraenkel-Mostowski sets and the NEW model of abstraction
Presented as part of a course in Tel-Aviv University at 12:00 on Sunday 28 May.
Presented by Murdoch J. Gabbay Mini-course Tel Aviv University Tel Aviv, Israel
Teaching and technical communication 6 talks
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Talk type: Teaching talk
Jamie’s musings and dubious advice
I dispense dubious life advice to PhD students participating in the SPLV summer school. I can’t guarantee that the advice is sound, but I can guarantee that every mistake listed is one which I have personally seen made (mostly by myself). If only I had known then what I know now, well, I’d probably have made a whole different collection of mistakes. This talk was dedicated to the memory of Gilles Dowek, who was a colleague, friend, and most pertinently to this talk, an inexhaustible source of wise advice.
Presented by Murdoch J. Gabbay Scottish Programming Languages and Verification Summer School 2025 (SPLV25) University of Edinburgh Edinburgh, Scotland
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Talk type: Course / lecture
Technical writing
A seminar given to local MSc students, on technical writing.
MSc course, Heriot-Watt University Edinburgh, UK
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Talk type: Teaching talk
On technical writing
A talk given to MSc students on (you guessed it) technical writing. We teach CS students to do technical things, but rarely do we focus on how to communicate about technical things. This can lead to problems later on, so I feel that this is an important lack and it is one which I am always happy to try and address.
Presented by Murdoch J. Gabbay MSc course, Heriot-Watt University Edinburgh, UK
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Talk type: Course / lecture
The laws of writing a good dissertation
I discuss technical writing for an audience of fourth year undergraduates, on Wednesday 22 October at 13:15. Oh, what are the laws of writing a good dissertation? Read the slides and find out, with examples!
Heriot-Watt University Edinburgh, UK
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Talk type: Teaching talk
Strange but true! Second year undergraduates chat all night online about logic course
A talk given as part of the Flexible Learning Talks series in Heriot-Watt University on Friday 11 January. Thanks to Nicole Cargill-Kipar.
Flexible Learning Talks series, Heriot-Watt University Edinburgh, UK
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Talk type: Teaching talk
Strange but true! 2nd year undergraduates chat all night online about logic course.
A talk given at the Flexible Learning Conference in Edinburgh at Heriot-Watt on May 18 at 14:05. I discuss my experience teaching the F22HO2 ‘formal specification’ course, and in particular the success of a message board I set up for the course, which attracted 1600 posts in five weeks and which remains active today. The video is the worst video in the history of worst videos. Sorry that I put the camera on the projector so it picked up the noise from the fan.
Flexible Learning Conference, Heriot-Watt University Edinburgh, UK
Theorem proving 1 talk
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Talk type: Workshop talk
Is there a quick way to get names and binding in some nominal sense up and running in Isabelle/HOL polymorphically on the type-class ‘type’, without having to rewrite the whole thing from scratch — yes or no?
Talk given at the Binding Challenges workshop in JAIST, Ishikawa, Japan. Thanks to Rene Vestergaard. We speculate on an interesting way in which it may be possible to encode syntax-with-binding in a higher-order logic based theorem-proving environment.
Binding Challenges workshop, Japan Advanced Institute of Science and Technology Ishikawa, Japan
Talks by title
Talks by type
Software
I hope you find the following software interesting and perhaps informative. I enjoyed writing it.
Programmable slide-rule emulator
Explore a two-sided scientific slide rule with configurable scales, classic instrument layouts, and guided lessons. Move the slide and cursor, then save your own instrument for offline use.
Aperiodic Life
Cellular automata on square, hexagonal, Einstein-hat, and Penrose tilings. You can change the rules, paint a starting pattern, and watch the evolution of one of a library of curated examples. Some of those examples are new, including some novel glider shapes.
3muQL model checker
A model checker for 3muQL, for checking whether a model satisfies specified logical properties. See the related Declarative Paxos paper.
zkFOL
An efficient implementation of the Arithmetisation of first-order logic paper, with thanks to Andrew Mendelsohn and Jeremy Ornelas.
Nominal Datatypes
Names, swapping, binding, and capture-avoiding programming, with examples for the lambda-calculus, System F, and EUTxO.
See also the August 2020 SPLV notes and recordings
Cairo integer types & testing
Archived Cairo 0.x tools from the earlier Python ecosystem: bitwise integers over field elements and property-based testing.
Scripts$
Small programs written when I remember that computers are supposed to save labour.
Writing
Notes on academic writing
Practical advice on papers, collaboration, and revision.
Treatment of an invited speaker
A satirical guide to hosting an academic visitor.
Essays & opinions
Longer reflections, provocations, and occasional miscellanea.
The Good, the Bad, and the Ugly
Things I approve of, disapprove of, or am shocked by.
Program hall of fame
Programs that changed my life.
The false market in degrees is ruining higher education
By Murdoch J. Gabbay. Published article in Times Higher Education (THE), 23 March 2022. (Archived copy.)
Karl Marx and the Blockchain
Devraj Basu and Murdoch J. Gabbay. Published book chapter in Media, Technology and Education in a Post-Truth Society, edited by Alex Grech (Emerald Publishing, 8 July 2021), pp. 225–241.
Recognition
“until just now I had thought the Booker Prize a rather trivial, showbizzy caper, but now I consider ... it a very serious, reliable indication of literary merit”
- Kingsly Amis, on winning the Booker prize, "Memoirs" page 325
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Recognised in
The Coinspondent · Friedemann Brenneis
Named
“Paper of the Year”
by The CoinspondentAuthors Devraj Basu and Murdoch J. Gabbay
In a 31 July 2020 article for the German-language Bitcoin publication The Coinspondent, Friedemann Brenneis called the then-new arXiv preprint
“the paper of the year”
, praising its intelligence, pointed formulations, and thought-provoking analysis. Thank you! More reviews like this, please.Featured paper
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Karl Marx and the Blockchain
Devraj Basu and Murdoch J. Gabbay · arXiv:2007.13346 (July 2020); later published in Media, Technology and Education in a Post-Truth Society, 225–241 (2021)
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Karl Marx and the Blockchain
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Recognised in
ACM SIGLOG · EATCS · EACSL · Kurt Gödel Society
Alonzo Church Award for Outstanding Contributions to Logic and Computation
Recipients Murdoch J. Gabbay and Andrew M. Pitts
Jointly awarded for introducing the theory of nominal representations: a mathematical framework for computation with atomic names, freshness, abstraction, binding, and scope.
Works named in the award citation
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A New Approach to Abstract Syntax with Variable Binding
Murdoch J. Gabbay and Andrew M. Pitts · Formal Aspects of Computing 13(3–5), 341–363 (2002)
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Nominal Logic, a First Order Theory of Names and Binding
Andrew M. Pitts · Information and Computation 186(2), 165–193 (2003)
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A New Approach to Abstract Syntax with Variable Binding
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Recognised in
ACM/IEEE Symposium on Logic in Computer Science
LICS Test-of-Time Award
Recipients Murdoch J. Gabbay and Andrew M. Pitts
Recognised twenty years after LICS 1999 for its lasting influence.
Recognised paper
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A New Approach to Abstract Syntax Involving Binders
Murdoch J. Gabbay and Andrew M. Pitts · Proceedings of LICS 1999, 214–224
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A New Approach to Abstract Syntax Involving Binders
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Recognised in
Principles and Practice of Declarative Programming
PPDP Most Influential Paper 10-Year Award
Recipients Maribel Fernández, Murdoch J. Gabbay, and Ian Mackie
Selected by the PPDP Steering Committee as the most influential paper from PPDP 2004, recognising its contribution to declarative programming and rewriting with names and binders.
Recognised paper
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Nominal Rewriting Systems
Maribel Fernández, Murdoch J. Gabbay, and Ian Mackie · Proceedings of PPDP 2004, 108–119
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Nominal Rewriting Systems
Miscellaneous
If you cite my work, please use Murdoch J. Gabbay rather than M. J. Gabbay: my brother Mike has the same initials.
Editorial work
- Journal of Financial Technology — Editor-in-chief
- Journal of Logic and Computation — Handling editor · deputy editor-in-chief
- Journal of Applied Logics — Editor
The Journal of Financial Technology is free and open-access, bringing together mathematics, statistics, computer science, finance, and law in fintech.
Food intolerance
I'm wheat-intolerant: I don't get an allergic reaction to wheat, but neither can I digest it, so it just sits in my digestion until it ... comes out, eventually.
Finding things to eat is not a problem, thank goodness, but I miss out on
convenience. I can't just "grab a sandwich"
.
I have to plan ahead, just in case.
- I can eat non-wheat carbohydrates like rice, potatoes, oats, barley, corn, and rye. Porridge, polenta, and pumpernickel are fine.
- Nuts, milk, and soya are not a problem.
- I can't eat pasta, pizza, cous-cous (cracked grain), naan, laffa flatbreads, doughnuts, baguettes, croissants (which is a real pity), sandwiches, or saitan (purified wheat gluten; that one is a big no-no).
- Potato gnocchi are fine; they contain a bit of wheat but it's generally not enough to bother me. Likewise, gravy thickened with a bit of wheat flour is fine. Funnily enough, I'm generally OK with a sinfully rich chocolate cake too: that's mostly sugar, fat, and chocolate, which I can digest very well.
I usually just ask for the "gluten-free"
option.
It's a simplification, but a useful one in practice.
Swedish Chef
For your convenience, my homepage is available in Swedish Chef (see Wikipedia and YouTube).
We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.