Skip to main content Accessibility help

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

  • Pierre-Louis Curien (a1) and Roberto Di Cosmo (a2)


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.



Hide All
Akama, Y. (1993) On Mints' reductions for ccc-Calculus. Typed lambda calculus and applications. LNCS 664, pp. 112. Springer-Verlag.
Barendregt, H. (1984) The lambda calculus; its syntax and semantics (revised edition). North Holland.
Breazu-Tannen, V. (1988) Combining algebra and higher order types. In: Proceedings of the symposium on logic in computer science (LICS), pp. 8290.
Breazu-Tannen, V. and Gallier, J. (1994) Polymorphic rewiting preserves algebraic confluence. Information and Computation. To appear.
Bruce, K., Di Cosmo, R. and Longo, G. (1992) Provable isomorphisms of types. Mathematical Structures in Computer Science, 2(2), 231247.
Cubric, D. (1992) On free CCC. Distributed on the types mailing list.
Curien, P.lL. and Ghelli, G. (1990) Confluence and decidability of βηtop≤ reduction on F≤. Information and Computation. To appear.
de Vrijer, R. C. (1987) Surjective pairing and strong normalization: two themes in λ-calculus. Ph.D. thesis, Universiteit van Amsterdam.
Dezani-Ciancaglini, M. (1976) Characterization of normal forms possessing an inverse in the λβη calculus. Theoretical Computer Science, 2, 323337.
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.
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. 462472. Lecture Notes in Computer Science, 820. Springer-Verlag.
Di Cosmo, Ro. and Kesner, D. (1994b) Simulating expansions without expansions. Mathematical Structures in Computer Science, 4, 148.
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).
Girard, J.-Y., Lafont, Y. and Taylor, P. (1990) Proofs and Types. Cambridge University Press.
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), 291342.
Jay, C. B. and Ghani, N. (1992) The Virtues of Eta-expansion. Tech. rept. ECS-LFCS-92-243. LFCS. University of Edimburgh.
Klop, J. W. (1980) Combinatory reduction systems. Mathematical Center Tracts, 27.
Lambek, J. and Scott, P. J. (1986) An Introduction to Higher Order Categorical Logic. Cambridge University Press.
Mints, G.A simple proof of the coherence theorem for cartesian closed categories. Bibliopolis. To appear.
Nipkow, T. (1990) A critical pair lemma for higher-order rewrite systems and its application to λ*. In: First Annual Workshop on Logical Frameworks.
Obtulowicz, A. (1987) Algebra of constructions I. The Word Problem for Partial Algebras. Information and Computation, 73(2), 129173.
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), 340376.
Pottinger, G. (1981) The Church Rosser Theorem for the Typed lambda-calculus with Surjective Pairing. Notre Dame Journal of Formal Logic, 22(3), 264268.
Tait, W. W. (1967) Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32.
Troelstra, A. S. (1986) Strong normalization for typed terms with surjective pairing. Notre Dame Journal of Formal Logic, 27(4).

Related content

Powered by UNSILO

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

  • Pierre-Louis Curien (a1) and Roberto Di Cosmo (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.

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

  • Pierre-Louis Curien (a1) and Roberto Di Cosmo (a2)
Submit a response


No Discussions have been published for this article.


Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *