Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-07-07T00:19:56.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependence and independence results for (impredicative) calculi of dependent types

Published online by Cambridge University Press:  04 March 2009

Thomas Streicher
Thomas Streicher
Affiliation:
Fakultät für Mathematik und Informatik, University of Passau, Postfach 2540, W-8390, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on a categorical semantics for impredicative calculi of dependent types we prove several dependence and independence results. Especially we prove that there exists a model where all usual syntactical concepts can be interpreted with only one exception: in the model the strong sum of a family of propositions indexed over a proposition need not be isomorphic to a proposition again.

The method of proof consists of restricting the set of propositions in the well known PERw model due to E. Moggi. The subsets of PERw considered in this paper are inspired by the subset ExpO of PERw introduced by Freyd et al.

Finally we show that a weak and a strong notion of sub-locally-cartesian-closed-category coincide under rather mild completeness conditions.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 2 , Issue 1 , March 1992 , pp. 29 - 54
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Coquand, Th. (1986) An analysis of Girard's paradox. In: Proc. 1st LICS Conf., Boston.Google Scholar
Ehrhard, T. (1988) Categorical semantics of constructions. In: Proc. 3rd LICS Conf., Edinburgh.Google Scholar
Ehrhard, T. (1989) Dictoses. In: Proc. 3rd Workshop on Category Theory in Computer Science. Springer Lecture Notes in Computer Science.Google Scholar
Freyd, P., Mulry, P., Rosolini, G. and Scott, D. (1989) Domains in PER, University of Pennsylvania, unpublished.Google Scholar
Hyland, M. and Pitts, A. (1989) The theory of constructions: categorical semantics and topostheoretic models. In: Proc. Conf. Categories in Computer Science and Logic. Contemp. Math. Amer. Math. Soc.Google Scholar
Meseguer, J. (1988) Universe models and initial model semantics for the second order polymorphic lambda calculus. Abstract in Abstracts of papers presented to A.M.S., 58 (9) Num. 4.Google Scholar
Martin-Löf, P. (1984) Intuitionistic type theory, notes by G. Sambin. Studies in Proof Theory.Bibliopolis, Naples.Google Scholar
Pitts, A. (1987) Polymorphism is set-theoretic, constructively. Proc. Conf. Category Theory and Computer Science, Edinburgh 1987. Springer Lecture Notes in Computer Science 283.Google Scholar
Rosolini, G. (1988) About modest sets. Preprint.Google Scholar
Seely, R. (1984) Locally Cartesian closed categories and type theory. In: Math. Proc. Cambridge Phil. Soc. 95.Google Scholar
Seely, R. (1987) Categorical semantics for higher order polymorphic lambda calculus. J. Symbolic Logic.Google Scholar
Streicher, T. (1988) Correctness and completeness of a categorical semantics of the calculus of constructions. PhD Thesis, University of Passau.Google Scholar
Seely, R.(1989) Independence results for calculi of dependent types. In: Proc. 3rd Workshop on Category Theory in Computer Science. Springer Lecture Notes in Computer Science.Google Scholar