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Simulating expansions without expansions

Published online by Cambridge University Press:  04 March 2009

Roberto Di Cosmo  and
Delia Kesner
Roberto Di Cosmo
Affiliation:
DMI-LIENS (CNRS URA 1347) Ecole Normale Supérieure - 45, Rue d'Ulm 75230 Paris France email: dicosmo@dmi.ens.fr
Delia Kesner
Affiliation:
INRIA Rocquencourt - Domaine de Voluceau, BP 105 - 78153 Le Chesnay Cedex, France and CNRS and LRI - Bât 490, Université de Paris-Sud - 91405 Orsay Cedex, France email: kesner@lri.lri.fr
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Abstract

We add extensional equalities for the functional and product types to the typed λ-calculus with, in addition to products and terminal object, sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus that stays confluent when allowing unbounded recursion. To do this, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual λ-calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non-extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence directly form our simulation technique without using the weak confluence property.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 4 , Issue 3 , September 1994 , pp. 315 - 362
Copyright
Copyright © Cambridge University Press 1994

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References

Akama, Y. (1993) On Mints’ reductions for ccc-calculus. In: Bezem, M. and Groote, J. (eds.) Typed Lambda Calculus and Applications. Springer-Verlag Lecture Notes in Computer Science 664 1–12CrossRefGoogle Scholar
Barendregt, H. (1984) The Lambda Calculus; Its syntax and Semantics (revised edition). North- Holland.Google Scholar
Curien, P.-L. and Di Cosmo, R. (1991) A confluent reduction system for the λ-calculus with surjective pairing and terminal object. In: Leach, , Monien, , AND Artalejo, (eds.) Intern. Conf. on Automata, Languages and Programming (ICALP). Springer-Verlag Lecture Notes in Computer Science 510 291–302CrossRefGoogle Scholar
Curry, H. and Feys, R. (1958) Combinatory Logic, volume 1. North-Holland.Google Scholar
Cubric, D. (1992) On free ccc. Distributed on the types mailing list.Google Scholar
Di Cosmo, R. and Kesner, D. (1993a) A confluent reduction for the extensional typed λ-calculus with pairs, sums, recursion and terminal object. In: Lingas, A. (ed.) Intern. Conf. on Automata, Languages and Programming (ICALP). Springer-Verlag Lecture Notes in Computer Science.CrossRefGoogle Scholar
Di Cosmo, R. and Kesner, D. (1993b) Simulating expansions without expansions. Technical Report LIENS-93–11/INRIA 1911, LIENS-DMI and INRIA.Google Scholar
Dougherty, D. J. (1990) Some reduction properties of a lambda calculus with coproducts and recursive types. Technical report, Wesleyan University, E-mail: ddougherty@eagle.wesleyan.edu.Google Scholar
Dougherty, D. J. (1993) Some lambda calculi with categorical sums and products. In: Proc. of the Fifth International Conference on Rewriting Techniques and Applications (RTA).CrossRefGoogle Scholar
Gallier, J. (1993) On the correspondence between proofs and λ-terms. (Available by anonymous ftp from ftp.cis.upenn.edu.)Google Scholar
Girard, J.-Y. (1972) Interprétation fonctionelle et élimination des coupures dans l’arithmétique d’ordre supérieure, Thèse de doctorat d’état, Université de Paris VII.Google Scholar
Girard, J.-Y., Lafont, Y. and Taylor, P. (1990) Proofs and Types, Cambridge University Press.Google Scholar
Huet, G. (1976) Résolution d’équations dans les langages d’ordre 1,2,…,ω, Thèse d’Etat, Université Paris VII.Google Scholar
Jay, C. B. (1992) Long βη normal forms and confluence (and its revised version). Technical Report ECS-LFCS-91–183, LFCS, University of Edimburgh.Google Scholar
Jay, C. B. and Ghani, N. (1992) The virtues of eta-expansion. Technical Report ECS-LFCS-92–243, LFCS, University of Edimburgh.Google Scholar
Klop, J. W. (1980) Combinatory reduction systems. Mathematical Center Tracts 27.Google Scholar
Lévy, J.-J. (1976) An algebraic interpretation of the λβκ-calculus and a labelled λ-calculus. Theoretical Computer Science 2 97–114CrossRefGoogle Scholar
Lambek, J. and Scott, P. J. (1986) An introduction to higher order categorical logic, Cambridge University Press.Google Scholar
Mints, G. (1977) Closed categories and the theory of proofs. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V.A. Steklova AN SSSR 68 83–114Google Scholar
Mints, G. (1979) Teorija categorii i teoria dokazatelstv. I. Aktualnye problemy logiki i metodologii nauky 252–278Google Scholar
Pottinger, G. (1981) The Church Rosser Theorem for the Typed lambda-calculus with Surjective Pairing. Notre Dame Journal of Formal Logic 22 (3) 264–268CrossRefGoogle Scholar
Prawitz, D. (1971) Ideas and results in proof theory. Proceedings of the 2nd Scandinavian Logic Symposium 235–307CrossRefGoogle 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) 340–376CrossRefGoogle Scholar
Troelstra, A. S. (1986) Strong normalization for typed terms with surjective pairing. Notre Dame Journal of Formal Logic 27 (4).CrossRefGoogle Scholar