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Categorical models for non-extensional λ-calculi and combinatory logic

Published online by Cambridge University Press:  04 March 2009

Simone Martini
Simone Martini
Affiliation:
Università di Pisa, Dipartimento do Informatica, Corso Italia, 40 I-56125 Pisa, Italy
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Abstract

The notions of weak Cartesian closed category and very weak CCC are introduced by dropping the extensionality (and the naturality) requirements in the adjunction defining the closed structure of a CCC. A number of specific examples of these categories are given. The weak notions are shown to be equivalent from both the semantic and syntactic standpoint to the typed non-extensional lambda-calculus and to the typed Combinatory Logic, extended with surjective pairs. Type-free models are characterized as reflexive objects in wCCCs. Finally, categorical models for the second-order non-extensional calculus are defined, by introducing a simple generalization of the notion of PL-category.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 2 , Issue 3 , September 1992 , pp. 327 - 357
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

Asperti, A. and Martini, S. (1991) Categorical models of polymorphism. Inf. and Comp. (to appear).Google Scholar
Bainbridge, E. S., Freyd, P. J., Scedrov, A. and Scott, P. J. (1990) Functorial polymorphism. In. Huet, . (ed), Logical Foundations of Functional Programming, Addison-Wesley.Google Scholar
Barendregt, H. (1984) The Lambda Calculus: Its Syntax and Semantics. Rev. ed., North-Holland.Google Scholar
Berry, G. (1980) On the definition of lambda-calculus models. INRIA-Roquencourt, Rapport 46.Google Scholar
Coquand, T. and Huet, G. (1985) Constructions: a hogher order proof system for mechanizing mathematics. EUROCAL 85, Springer Lecture Notes in Computer Science 203 151–184.CrossRefGoogle Scholar
Coquand, T. and Ehrhardt, T. (1987) An equational presentation of higher order logic. In: Pitts, et al. (eds), Category Th. and comp. Science. Edinburgh, Springer Lecture Notes in Computer Science 283.CrossRefGoogle Scholar
Curien, P. L. (1986) Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, London.Google Scholar
Girard, J. Y. (1972) Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'Etat, Paris.Google Scholar
Hayashi, S. (1985) Adjunction of semifunctors: categorical structures in non-extensional lambda calcus. Theor. Comput. Sci. 41(1) 95–104.CrossRefGoogle Scholar
Hindley, J. R. and Seldin, J. P. (1986) Introduction to Combinators and λ-calculus. Cambridge University Press, Cambridge.Google Scholar
Kainen, (1971) Weak adjoint functors. Math. Z. 122 1–9.CrossRefGoogle Scholar
Koymans, K. (1982) Models of the lambda calculus. Inf. and Control. 52 306–332.CrossRefGoogle Scholar
Lambek, J. (1980) From lambda-calculus to Cartesian closed categories. In: Hindly, and Seldin, (eds), Essays in Combinatory Logic Lambda-calculus and Formalism. Academic Press.Google Scholar
Lambek, J. (1985) Cartesian closed categories and typed lambda-calculi. In Cousineay, et al. (eds), Combinators and Functional Programming Languages, Springer Lecture Notes in Computer Science 242 136–175.CrossRefGoogle Scholar
Lambek, J. and Scott, P. J. (1986) Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge.Google Scholar
Longo, G. and Moggi, E. (1990) A category-theoretic characterization of functional completeness. Theor. Comput. Sci. 70 193–211.CrossRefGoogle Scholar
MacQueen, D., Plotkin, G. and Sethi, R. (1986) An ideal model for recursive polymorphic types. Information and Control 71 95–130.CrossRefGoogle Scholar
Martini, S. (1987) An interval model for second order lambda calculus. In: Pitts, et al. (eds), Category Th. and Comp. science, Edinburgh, Springer Lecture Notes in Computer Science 283 219–237.CrossRefGoogle Scholar
Martini, S. (1988) Modelli non estensionali del ploimorfismo in programmazione funzionale. Tesi di Dottorato, Università di Pisa.Google Scholar
Meseguer, J. (1989) Relating models of polymorphism. Popl 1989, ACM 228–241.CrossRefGoogle Scholar
Moggi, E. (1987) Interpretation of second-order lambda-calculus in categories. Unpublished notes.Google Scholar
Mitchell, J. and Scott, P. J. (1989) Typed lambda models and Cartesian closed categories. Contemp. Math. 92 301–316.CrossRefGoogle Scholar
Obtulowicz, A. (1987) Algebra of constructions I. Inf. and Comp. 73 129–173.CrossRefGoogle Scholar
Pitts, A. (1987) Polymorphism is set-theoretic, constructiveluy. In: Pitts, et al. (eds), Category Th. and Comp. Science., Edinburgh, Springer Lecture Notes in Computer Science 283.CrossRefGoogle Scholar
Poigné, A. (1986) On specification theories and models with higher types. Information and Control 68 1–46.CrossRefGoogle Scholar
Reynolds, J. (1974) Towards a theory of type structures. Colloque sur la Programmation, Springer Lecture Notes in Computer Science 19 408–425.CrossRefGoogle Scholar
Reynolds, J. (1985) Three approaches to type structure. Springer Lecture Notes in Computer Science 185 145–146.Google Scholar
Scott, D. (1980) Relating theories of the lambda-calculus. In: Hindley, and Seldin, (eds), Essays in Combinatory Logic Lambda-calculus and Formalism. Academic Press, 1984.Google Scholar
Seely, R. A. G. (1987) Categorical semantics for higher order polymorphic lambda calculus. JSL 52 (4) 969–989.Google Scholar
Troeslstra, A. (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer Lecture Notes in Mathematics 344.CrossRefGoogle Scholar
Wiweger, A. (1984) Pre-adjunction and λ-algebraic theories. Colloq. Math. 48(2) 153–165.CrossRefGoogle Scholar