Abstract
This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived. Its axioms express key properties of these primitives, which are satisfied by the FM-sets model of syntax introduced in [11]. Nominal Logic serves as a vehicle for making two general points. Firstly, nameswapping has much nicer logical properties than more general forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α-conversion. Secondly, it is useful for the practice of operational semantics to make explicit the equivariance property of assertions about syntax — namely that their validity is invariant under name-swapping.
Research funded by UK EPSRC grant GR/R07615.
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References
M. J. Beeson. Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebeite. Springer-Verlag, Berlin, 1985.
L. Caires and L. Cardelli. A spatial logic for concurrency. Draft of 11 April, 2001.
L. Cardelli and A. D. Gordon. Logical properties of name restriction. In S. Abramsky, editor, Typed Lambda Calculus and Applications, 5th International Conference, volume 2044 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2001.
The Coq proof assistant. Institut National de Recherche en Informatique et en Automatique, France. http://coq.inria.fr/
N. G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indag. Math., 34:381–392, 1972.
J. Despeyroux, F. Pfenning, and C. Schurmann. Primitive recursion for higherorder abstract syntax. In Typed Lambda Calculus and Applications, 3rd International Conference, volume 1210 of Lecture Notes in Computer Science, pages 147–163. Springer-Verlag, Berlin, 1997.
G. Dowek, T. Hardin, and C. Kirchner. A completeness theorem for an extension of first-order logic with binders. In S. Ambler, R. Crole, and A. Momigliano, editors, Mechanized Reasoning about Languages with Variable Binding (MERLIN 20001),Proceedings of a Workshop held in conjunction with the International Joint Conference on Automated Reasoning, IJCAR 2001, Siena, June 2001, Department of Computer Science Technical Report 2001/26, pages 49–63. University of Leicester, 2001.
P. J. Freyd. The axiom of choice. Journal of Pure and Applied Algebra, 19:103–125, 1980.
M. J. Gabbay. AThe ory of Inductive Definitions with ?-Equivalence: Semantics, Implementation, Programming Language. PhD thesis, Cambridge University, 2000.
M. J. Gabbay and A. M. Pitts. A new approach to abstract syntax involving binders. In 14th Annual Symposium on Logic in Computer Science, pages 214–224. IEEE Computer Society Press, Washington, 1999.
M. J. Gabbay and A. M. Pitts. A new approach to abstract syntax with variable binding. Formal Aspects of Computing, 2001. Special issue in honour of Rod Burstall. To appear.
A. D. Gordon and T. Melham. Five axioms of alpha-conversion. In Theorem Proving in Higher Order Logics: 9th Interational Conference, TPHOLs’96, volume 1125 of Lecture Notes in Computer Science, pages 173–191. Springer-Verlag, Berlin, 1996.
C. A. Gunter. Semantics of Programming Languages: Structures and Techniques. Foundations of Computing. MIT Press, 1992.
K. Honda. Elementary structures in process theory (1): Sets with renaming. Mathematical Structures in Computer Science, 10:617–663, 2000.
F. Honsell, M. Miculan, and I. Scagnetto. An axiomatic approach to metareasoning on systems in higher-order abstract syntax. In Twenty-Eighth International Colloquium on Automata, Languages and Programming, ICALP 2001, Crete, Greece, July 2001, Proceedings, Lecture Notes in Computer Science. Springer-Verlag, Heidelberg, 2001.
B. Huppert. Endliche Gruppen I, volume 134 of Grundlehren Math. Wiss. Springer, Berlin, 1967.
T. J. Jech. About the axiom of choice. In J. Barwise, editor, Handbook of Mathematical Logic, pages 345–370. North-Holland, 1977.
M. Makkai and G. E. Reyes. First Order Categorical Logic, volume 611 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1977.
J. McKinna and R. Pollack. Some type theory and lambda calculus formalised. To appear in Journal of Automated Reasoning, Special Issue on Formalised Mathematical Theories (F. Pfenning, Ed.), 1998.
M. Miculan. Developing (meta)theory of lambda-calculus in the theory of contexts. In S. Ambler, R. Crole, and A. Momigliano, editors, Mechanized Reasoning about Languages with Variable Binding (MERLIN 20001), Proceedings of aWorkshop held in conjunction with the International Joint Conference on Automated Reasoning, IJCAR 2001, Siena, June 2001, Department of Computer Science Technical Report 2001/26, pages 65–81. University of Leicester, 2001.
L. C. Paulson. Isabelle: AGeneric Theorem Prover, volume 828 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1994.
F. Pfenning and C. Elliott. Higher-order abstract syntax. In Proc. ACMSIGPLAN Conference on Programming Language Design and Implementation, pages 199–208. ACM Press, 1988.
A. M. Pitts. Categorical logic. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures, chapter 2. Oxford University Press, 2000.
A. M. Pitts and M. J. Gabbay. A metalanguage for programming with bound names modulo renaming. In R. Backhouse and J. N. Oliveira, editors, Mathematics of Program Construction. 5th International Conference, MPC2000, Ponte de Lima, Portugal, July 2000. Proceedings, volume 1837 of Lecture Notes in Computer Science, pages 230–255. Springer-Verlag, Heidelberg, 2000.
C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie-Mellon University, 2000.
D. S. Scott. Identity and existence in intuitionistic logic. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, Proceedings, Durham 1977, volume 753 of Lecture Notes in Mathematics, pages 660–696. Springer-Verlag, Berlin, 1979.
R. Vestergaard and J. Brotherson. A formalised first-order confluence proof for the λ-calculus using one-sorted variable names. In A. Middeldorp, editor, Rewriting Techniques and Applications, 12th International Conference, RTA 2001, Utrecht, The Netherlands, May 2001, Proceedings, volume 2051 of Lecture Notes in Computer Science, pages 306–321. Springer-Verlag, Heidelberg, 2001.
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Pitts, A.M. (2001). Nominal Logic: A First Order Theory of Names and Binding. In: Kobayashi, N., Pierce, B.C. (eds) Theoretical Aspects of Computer Software. TACS 2001. Lecture Notes in Computer Science, vol 2215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45500-0_11
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