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Semantics of the second order lambda calculus†

Published online by Cambridge University Press:  04 March 2009

Bart Jacobs
Bart Jacobs
Affiliation:
Dep. Comp. Sc., Toernooiveld, 6525 ED Nijmegen, The Netherlands
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Abstract

In the literature ther are two main notins of model for the second order λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely. Here we generalise Seely's notion, using semifunctors and semi-adjunctions from Hayashi, and introduce λ2-algebras, λη2-algebras, λ2-models and λη-models, similarly to the untyped λ-calculus. Non-extensional abstraction of both term and type variables is described by semi-adjunctions (essentially as in Martini's thesis). We show that also for second order sume, the β-(and commutation!) conversions correspond to semi-functoriality and that the (additional) η-conversion corresponds to ordinary fuctioriality.

In the above framework various examples – well known ones and variations – are described. Also, we determine the place of the BMM-models; an earlier version of the latter has been reported by Jacobs.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 3 , November 1991 , pp. 327 - 360
Copyright
Copyright © Cambridge University Press 1991

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References

Asperti, A. and longo, G. (1991) Applied Category Theory, book in prepartion.Google Scholar
Barendregt, H. P. (1984) The Lambda Calculus. Its Syntax and Semanties (revised second edition), North-Holland, Amsterdam.Google Scholar
Barendregt, H. P. and Rezus, A. (1983) Semanties for classical AUTOMATH and related systems. Inf. & Contr. 59 127–47.Google Scholar
Bruce, K. B. and Meyer, A. R. (1984) The semantics of second-order lambda calculus. In: Kahn, G., MacQueen, D. B. and Plotkin, G., eds, Proc. Conf. on Semanties of Data Types, Sophia-Antipolis, 06 1984, Springer Lecture Notes in Computer Science 173 131–144.CrossRefGoogle Scholar
Bruce, K. B., Meyer, A. R. and Mitchell, J. C. (1990) The semantics of second-order lambda calculus. Inf. & Comp. 85 76–134.CrossRefGoogle Scholar
Coquand, Th. and Ehrhard, Th. (1987) An equational presentaion of higher order logic. In: Pitt, et al. (1987) 40–56.Google Scholar
Coquand, Th., Gunter, C. and Winskel, G. (1989) Domain theoretic models of polymorphism. Inf. & Comp. 81 123–67.CrossRefGoogle Scholar
Ehrhard, Th. (1988) A categorical semantics of constructions. In: Proc. Third Ann. Symp. on Logic in Computer Science, Edinburgh, 07 1988, 264–73.Google Scholar
Friedman, H. (1975) Equality between founctionals. In: Parikh, R., ed., Logic Colloquium, Symposium on Logic held at Boston, 1972–1973, Springer Lecture Notes in Mathematics 453.Google Scholar
Girard, J.-Y. (1972) Interprération fonctinelle et éLimination des coupure-s de l'arithmétique d ordre supérieur. Thèse d'État, Paris VII.Google Scholar
Girard, J.-Y. (1986) The system F of variable types, fifteen years later. Th. Comp. Sci. 22 159–92.Google Scholar
Girard, J.-Y., Lafont, Y. and Taylor, P. (1989) Proofs and types. Camb. Univ. Press Tracts in Th. Comp. Sci. 7.Google Scholar
Goldblatt, R. (1984) Topoi. The categorial analysis of Logic (revised second edition), North-Holland, Amsterdam.Google Scholar
Hayashi, S. (1985) Adjunction of semifuntors: categorical structures in non-extensional lambdacalculus, Th. Comp. Sci. 41 95–104.Google Scholar
Hoofman, R. (1990) The linear decomposition of λ2-models, manuscript, Dep. comp. Sc. Utrecht Univ., The Netherlands.Google Scholar
Hyland, J. M. E. (1989) A small complets category. Ann. Pure Appl. Log. 40 135–65.Google Scholar
Jacobs, B. (1989) On the semanties of the second order λ-calculus: from Bruce-Meyer-Mitechell models to hyperdoctrine models and vice-versa. In: Pitt, D. H. et al. , eds, Category Theory and Computer Science, Springer Lecture Notes in Computer Science 389 198–212.CrossRefGoogle Scholar
Jacobs, B., Margaria, I. and Zacchi, M. (1989) Filter models with polymorphic Types.Google Scholar
Koymans, K. (1982) Models of the lambda calculus. Inf. & Contr. 52 306–32.Google Scholar
Longo, G. and Moggi, E. (1988) Constructive natural deduction and its ‘modest’ interpretation. Report SC-88–131, carnegie Mellon Univ. Also in: MSCS, no. 2, vol. 1.Google Scholar
MacLane, S. (1972) Categories for the working Mathematician, Springer.Google Scholar
MacQueen, K., Sethi, R. and Plotkin, G. (1986) An ideal model for recursive polymorphis types. Inf. & Contr. 71 95–130.CrossRefGoogle Scholar
Martini, S. (1987) An interval model for second order lambda calculus. In: Pitt, et al. (1987) 219–37.Google Scholar
Martini, S. (1988) Modelli non estensionali del polimorfismo in programmazine funzinale. PhD thesis, University of Pisa.Google Scholar
Mitchell, J. C. (1987) Polymorphism type inference and containment. Inf. & Comp. 76 (2/3), 211–49.Google Scholar
Pitt, A. M. (1987) Polymorphism is set theoretic, constructively. In: Pitt, et al. (1987) 12–39.Google Scholar
Pitt, D. H.Poigné, A. and Rydeheard, D., eds. (1987) Category Theory and Computer Science, Springer Lecture Notes in Computer Science 283.CrossRefGoogle Scholar
Prawits, D. (1975) Natural Deduction, Almqvist & Wiksell, Stockholm.Google Scholar
Reynolds, J. (1974) Towards a Theory of type structure. In: Paris Colloquium on Programming, Springer Lecture Notes in Computer Science 19 408–25.CrossRefGoogle Scholar
Scott, D. S. (1976) Data Types as lattices, SIAM J. comp. 5 522–87.Google Scholar
Seely, R. A. G. (1983) Hyperdoctrines, natural deduction and the Beck condition. Zeitschr. Math. Log. Grundl. Math. 29 33–48.Google Scholar
Seely, R. A. G. (1987). Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Log. 52 969–89.Google Scholar
Streicher, Th. (1988) Correctiness and completeness of a categorical semantics of the calculus of constructions. Thesis, University of Passau.Google Scholar