Murdoch J. Gabbay
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Abstract
Call a semantics for a language with variables absolute when variables map to fixed entities in the denotation. That is, a semantics is absolute when the denotation of a variable a is a copy of itself in the denotation.
We give a trio of lattice-based, sets-based, and algebraic absolute semantics to first-order logic. Possibly open predicates are directly interpreted as lattice elements/sets/algebra elements, subject to suitable interpretations of the connectives and quantifiers. In particular, universal quantification ∀a.φ is interpreted using a new notion of “fresh-finite” limit Λ #a ⟦Φ⟧ and using a novel dual to substitution.
The interest in this semantics is partly in the nontrivial and beautiful technical details, which also offer certain advantages over existing semantics. Also, the fact that such semantics exist at all suggests a new way of looking at variables and the foundations of logic and computation, which may be well suited to the demands of modern computer science.
- Haskell B. Curry and R. Feys. 1958. Combinatory Logic, vol. I, North Holland.Google Scholar
- Wil Dekkers, Martin Bunder, and Henk Barendregt. 1998. Completeness of Two Systems of Illative Combinatory Logic for First-Order Propositional and Predicate Calculus. Archive für Mathematische Logik 37, 327--341.Google ScholarCross Ref
- Gilles Dowek and Murdoch J. Gabbay. 2010. Permissive Nominal Logic. Proceedings of the 12th International ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2010) ACM, New York, NY, 165--176. Google ScholarDigital Library
- Gilles Dowek and Murdoch J. Gabbay. 2012a. Nominal Semantics for Predicate Logic: Algebras, Substitution, Quantifiers, and Limits, Proceedings of the 9th Italian Convention on Computational Logic (CILC’12), CEUR Workshop Proceedings, Vol. 857.Google Scholar
- Gilles Dowek and Murdoch J. Gabbay. 2012b. Permissive Nominal Logic. Transactions on Computational Logic 13, 3. Google ScholarDigital Library
- B. A. Davey and Hilary A. Priestley. 2002. Introduction to Lattices and Order (2nd ed.). Cambridge University Press, New York, NY.Google Scholar
- Maribel Fernández and Murdoch J. Gabbay. 2007. Nominal rewriting.Information and Computation 205, 6, 917--965. Google ScholarDigital Library
- Maribel Fernández, Murdoch J. Gabbay, and Ian Mackie. 2004. Nominal Rewriting Systems. Proceedings of the 6th ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP’04), ACM, New York, 108--119. Google ScholarDigital Library
- Marcelo Fiore and Chung-Kil Hur. 2008. Term Equational Systems and Logics. Electronic Notes in Theoretical Computer Science 218, 171--192. Google ScholarDigital Library
- Marcelo Fiore and Chung-Kil Hur. 2010. Second-Order Equational Logic. Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL’10). Lecture Notes in Computer Science, Springer. Berlin. Google ScholarDigital Library
- Kit Fine. 1985. Reasoning with Arbitrary Objects, Blackwell.Google Scholar
- Henrik Forssell. 2007. First-Order Logical Duality. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA.Google Scholar
- Murdoch J. Gabbay. 2001. A Theory of Inductive Definitions with Alpha-Equivalence. Ph.D. thesis, University of Cambridge, UK.Google Scholar
- Murdoch J. Gabbay. 2007. A General Mathematics of Names. Information and Computation 205, 7, 982--1011. Google ScholarDigital Library
- Murdoch J. Gabbay. 2009a. Nominal Algebra and the HSP Theorem. Journal of Logic and Computation 19, 2, 341--367. Google ScholarDigital Library
- Murdoch J. Gabbay. 2009b. A Study of Substitution, Using Nominal Techniques and Fraenkel-Mostowski Sets, Theoretical Computer Science 410, 12--13, 1159--1189. Google ScholarDigital Library
- Murdoch J. Gabbay. 2011. Foundations of Nominal Techniques: Logic and Semantics of Variables in Abstract Syntax. Bulletin of Symbolic Logic 17, 161--229.Google ScholarCross Ref
- Murdoch J. Gabbay. 2012. Unity in Nominal Equational Reasoning: The Algebra of Equality on Nominal Sets, Journal of Applied Logic 10, 199--217.Google ScholarCross Ref
- Murdoch J. Gabbay. 2013. Nominal Terms and Nominal Logics: From Foundations to Meta-Mathematics, In Handbook of Philosophical Logic, vol. 17, Kluwer.Google Scholar
- Murdoch J. Gabbay. 2014. Stone Duality for First-Order Logic: A Nominal Approach, HOWARD-60. A Festschrift on the Occasion of Howard Barringer’s 60th Birthday, Easychair Books.Google Scholar
- Murdoch J. Gabbay and Vincenzo Ciancia. 2011. Freshness and Name-Restriction in Sets of Traces with Names, Foundations of software science and computation structures, 14th International Conference (FOSSACS’11), Lecture Notes in Computer Science, Vol. 6604, Springer, Berlin, 365--380. Google ScholarDigital Library
- Murdoch J. Gabbay and Michael Gabbay. 2008. Substitution for Fraenkel-Mostowski Foundations. In Proceedings of the 2008 AISB Symposium on Computing and Philosophy, 65--72.Google Scholar
- Michael J. Gabbay and Murdoch J. Gabbay. 2010. A Simple Class of Kripke-Style Models in which Logic and Computation have Equal Standing. In International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR 2010). Google ScholarDigital Library
- Murdoch J. Gabbay and Michael J. Gabbay. 2016. Representation and Duality of the Untyped Lambda-Calculus in Nominal Lattice and Topological Semantics, with a Proof of Topological Completeness, In Annals of Pure and Applied Logic. See also arXiv preprint 1305.5968.Google Scholar
- Murdoch J. Gabbay and Martin Hofmann. 2008. Nominal Renaming Sets. In Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’08). Springer, Berlin, 158--173. Google ScholarDigital Library
- Murdoch J. Gabbay, Tadeusz Litak, and Daniela Petrişan. 2011. Stone Duality for Nominal Boolean Algebras with NEW. In Proceedings of the 4th International Conference on Algebra and Coalgebra in Computer Science (CALCO’11), Lecture Notes in Computer Science, Vol. 6859, Springer, Berlin, 192--207. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2006a. Capture-Avoiding Substitution as a Nominal Algebra. In Proceedings of the 3rd International Colloquium on Theoretical Aspects of Computing (ICTAC’06). Lecture Notes in Computer Science, Vol. 4281, Springer, Berlin, 198--212. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2006b. One-and-a-Halfth-Order Logic. In Proceedings of the 8th ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP’06), ACM, New York, NY, 189--200. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2007. A Formal Calculus for Informal Equality with Binding. In 14th Workshop on Logic, Language, Information and Computation (WoLLIC’07). Lecture Notes in Computer Science, Vol. 4576, Springer, Berlin, 162--176. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2008a. Capture-Avoiding Substitution as a Nominal Algebra, Formal Aspects of Computing 20, 4--5, 451--479.Google ScholarCross Ref
- Murdoch J. Gabbay and Aad Mathijssen. 2008b. The Lambda-Calculus is Nominal Algebraic, Reasoning in simple type theory: Festschrift in Honour of Peter B. Andrews on his 70th Birthday, Christoph Benzmüller, Chad Brown, Jörg Siekmann, and Rick Statman, (eds.), Studies in Logic and the Foundations of Mathematics, IFCoLog.Google Scholar
- Murdoch J. Gabbay and Aad Mathijssen. 2008c. One-and-a-Halfth-Order Logic. Journal of Logic and Computation 18, 4, 521--562. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2009. Nominal Universal Algebra: Equational Logic with Names and Binding, Journal of Logic and Computation 19, 6, 1455--1508. Google ScholarDigital Library
- Murdoch J. Gabbay and Aad Mathijssen. 2010. A Nominal Axiomatisation of the Lambda-Calculus, Journal of Logic and Computation20, 2, 501--531. Google ScholarDigital Library
- Murdoch J. Gabbay and Dominic P. Mulligan. 2011. Nominal Henkin Semantics: Simply-Typed Lambda-Calculus Models in Nominal Sets. Proceedings of the 6th International Workshop on Logical Frameworks and Meta-Languages (LFMTP’11), EPTCS, Vol. 71, 58--75.Google Scholar
- Robert Goldblatt. 1989. Varieties of Complex Algebras. Annals of Pure and Applied Logic 44 3, 173--242.Google Scholar
- Murdoch J. Gabbay and Andrew M. Pitts. 2001. A New Approach to Abstract Syntax with Variable Binding, Formal Aspects of Computing 13, no. 3--5, 341--363.Google Scholar
- Mariana Haim. 2000. Duality for Lattices with Operators: A Modal Logic Approach, Ph.D. thesis, University of Amsterdam, Amsterdam, The Netherlands.Google Scholar
- Paul R. Halmos. 2006. Algebraic Logic, AMS Chelsea Publishing, New York, NY.Google Scholar
- Leon Henkin, J. Donald Monk, and Alfred Tarski. 1971. Cylindric Algebras, North Holland, 1971 and 1985, Parts I and II.Google Scholar
- C. Norris Ip and David L. Dill. 1996. Better Verification through Symmetry. Formal Methods in System Design 9, 41--75. Google ScholarDigital Library
- Peter T. Johnstone. 2003. Sketches of an Elephant: A Topos Theory Compendium, Oxford Logic Guides, Vols. 43 and 44, Oxford University Press, New York, NY.Google Scholar
- Bjarni Jonnson and Alfred Tarski. 1952. Boolean Algebras with Operators, American Journal of Mathematics 74, 1, 127--162.Google ScholarCross Ref
- Max Kelly. 1982. Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series, No. 64, Cambridge University Press, New York, NY.Google Scholar
- Edward Keenan and Dag Westerståhl. 1996. Generalized Quantifiers in Linguistics and Logic, Handbook of Logic and Language, J. Van Benthem and A. Ter Meulen( eds.), Elsevier, 837--894.Google Scholar
- Saunders Mac Lane and Ieke Moerdijk. 1992. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext, Springer.Google Scholar
- William McCune, Robert Veroff, Branden Fitelson, Kenneth Harris, Andrew Feist, and Larry Wos. 2002. Short Single Axioms for Boolean Algebra. Journal of Automated Reasoning 1, 29, 1--16. Google ScholarDigital Library
- Donald Monk. 2000. An Introduction to Cylindric Set Algebras. Logic Journal of the IGPL 8, 4, 451--492.Google ScholarCross Ref
- Alan J. Perlis. 1982. Epigrams on Programming. Retrieved May 6, 2016 from www.cs.yale.edu/quotes.html, 7--13.Google Scholar
- Andrew M. Pitts. 2013. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press, New York, NY. Google ScholarCross Ref
- Moses Schönfinkel. 1967. On the Building Blocks of Mathematical Logic. In Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Jean van Heijenoort, (ed.), Harvard University Press, translated from the German by Stefan Bauer-Mengelberg.Google Scholar
- Mark R. Shinwell and Andrew M. Pitts. 2005. Fresh Objective CAML User Manual, Technical Report UCAM-CL-TR-621, University of Cambridge, Cambridge, UK.Google Scholar
- Alfred Tarski. 1944. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy and Phenomenological Research 4.Google Scholar
- Alfred Tarski and Steven Givant. 1987. A Formalization of Set Theory without Variables, Vol. 41, American Mathematical Society Colloquium Publications.Google Scholar
- Dirk van Dalen. 1994. Logic and Structure(3rd ed.) Universitext, Springer.Google Scholar
- Dag Westerståhl. 1989. Quantifiers in Formal and Natural Languages. In Handbook of Philosophical Logic, Synthèse, Vol. 4, Reidel, Dordrecht, The Netherlands, 1--131.Google ScholarCross Ref
Index Terms
- Semantics Out of Context: Nominal Absolute Denotations for First-Order Logic and Computation
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