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A confluent reduction for the λ-calculus with surjective pairing and terminal object

Part of: JFP Research Articles

Published online by Cambridge University Press:  07 November 2008

Pierre-Louis Curien  and
Roberto Di Cosmo
Pierre-Louis Curien
Affiliation:
LIENS (CNRS URA 1327) - DMI
Roberto Di Cosmo
Affiliation:
LIENS (CNRS URA 1327) - DMI, and Dipartimento di Scienze dell'Informazione, Pisa, Italy
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Abstract

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We exhibit confluent and effectively weakly normalizing (thus decidable) rewriting systems for the full equational theory underlying cartesian closed categories, and for polymorphic extensions of it. The λ-calculus extended with surjective pairing has been well-studied in the last two decades. It is not confluent in the untyped case, and confluent in the typed case. But to the best of our knowledge the present work is the first treatment of the lambda calculus extended with surjective pairing and terminal object via a confluent rewriting system, and is the first solution to the decidability problem of the full equational theory of Cartesian Closed Categories extended with polymorphic types. Our approach yields conservativity results as well. In separate papers we apply our results to the study of provable type isomorphisms, and to the decidability of equality in a typed λ-calculus with subtyping.

Type
Articles
Information
Journal of Functional Programming , Volume 6 , Issue 2 , March 1996 , pp. 299 - 327
Copyright
Copyright © Cambridge University Press 1996

References

Akama, Y. (1993) On Mints' reductions for ccc-Calculus. Typed lambda calculus and applications. LNCS 664, pp. 1–12. Springer-Verlag.Google Scholar
Barendregt, H. (1984) The lambda calculus; its syntax and semantics (revised edition). North Holland.Google Scholar
Breazu-Tannen, V. (1988) Combining algebra and higher order types. In: Proceedings of the symposium on logic in computer science (LICS), pp. 82–90.CrossRefGoogle Scholar
Breazu-Tannen, V. and Gallier, J. (1994) Polymorphic rewiting preserves algebraic confluence. Information and Computation. To appear.CrossRefGoogle Scholar
Bruce, K., Di Cosmo, R. and Longo, G. (1992) Provable isomorphisms of types. Mathematical Structures in Computer Science, 2(2), 231–247.CrossRefGoogle Scholar
Cubric, D. (1992) On free CCC. Distributed on the types mailing list.Google Scholar
Curien, P.lL. and Ghelli, G. (1990) Confluence and decidability of βηtop≤ reduction on F≤. Information and Computation. To appear.Google Scholar
de Vrijer, R. C. (1987) Surjective pairing and strong normalization: two themes in λ-calculus. Ph.D. thesis, Universiteit van Amsterdam.Google Scholar
Dezani-Ciancaglini, M. (1976) Characterization of normal forms possessing an inverse in the λβη calculus. Theoretical Computer Science, 2, 323–337.CrossRefGoogle Scholar
Di Cosmo, R. (1994) Second order isomorphic types. A proof theoretic study on second order λ-calculus with surjective pairing and terminal object. Information and Computation. To appear.CrossRefGoogle Scholar
Di Cosmo, R. and Kesner, D. (1994 a) Combining first order algebraic rewriting systems, recursion and extensional lambda calculi. In: Abiteboul, S. and Shamir, E. (eds), International Conference on Automata, Languages and Programming (ICALP), pp. 462–472. Lecture Notes in Computer Science, 820. Springer-Verlag.CrossRefGoogle Scholar
Di Cosmo, Ro. and Kesner, D. (1994b) Simulating expansions without expansions. Mathematical Structures in Computer Science, 4, 1–48.CrossRefGoogle Scholar
Dougherty, D. J. (1993) Some lambda calculi with categorical sums and products. In: Proceedings of the Fifth International Conference on Rewriting Techniques and Applications (RTA).CrossRefGoogle Scholar
Girard, J.-Y., Lafont, Y. and Taylor, P. (1990) Proofs and Types. Cambridge University Press.Google Scholar
Hardin, T. (1989) Confluence results for the pure strong categorical logic C.C.L.; λ-calculi as subsystems of C.C.L. Theoretical Computer Science, 65(2), 291–342.CrossRefGoogle Scholar
Jay, C. B. and Ghani, N. (1992) The Virtues of Eta-expansion. Tech. rept. ECS-LFCS-92-243. LFCS. University of Edimburgh.Google Scholar
Klop, J. W. (1980) Combinatory reduction systems. Mathematical Center Tracts, 27.Google Scholar
Lambek, J. and Scott, P. J. (1986) An Introduction to Higher Order Categorical Logic. Cambridge University Press.Google Scholar
Mints, G.A simple proof of the coherence theorem for cartesian closed categories. Bibliopolis. To appear.Google Scholar
Nipkow, T. (1990) A critical pair lemma for higher-order rewrite systems and its application to λ*. In: First Annual Workshop on Logical Frameworks.Google Scholar
Obtulowicz, A. (1987) Algebra of constructions I. The Word Problem for Partial Algebras. Information and Computation, 73(2), 129–173.CrossRefGoogle Scholar
Poigné, A. and Voss, J. (1987) On the implementation of abstract data types by programming language constructs. Journal of Computer and System Science, 34(2–3), 340–376.CrossRefGoogle Scholar
Pottinger, G. (1981) The Church Rosser Theorem for the Typed lambda-calculus with Surjective Pairing. Notre Dame Journal of Formal Logic, 22(3), 264–268.CrossRefGoogle Scholar
Tait, W. W. (1967) Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32.CrossRefGoogle Scholar
Troelstra, A. S. (1986) Strong normalization for typed terms with surjective pairing. Notre Dame Journal of Formal Logic, 27(4).CrossRefGoogle Scholar
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