Abstract
This work presents an extension of system AF 2 to allow the use of infinite data types. We extend the logic with inductive and coinductive types, and show that the “programming method” is still correct. We prove propositions about the normalization of typed terms. Moreover, since we only use the pure λ-calculus to represent data types, we prove uniqueness of the representation of data up to Böhm tree equivalence. We also study the problem of equality on infinite structures and we introduce a new approach, different from the coinduction scheme, and which suits our system better.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
H. P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, revised edition, 1984.
C. Böhm. Alcune proprieta delle forme βη-normali nel λκ-calculus. Pubblicazioni 696, Instituto per le Applicazioni del Calcolo, Roma, 1968.
T. Coquand and G. Huet. The calculus of construction. In Information and Computation, pages 241–262, 1988.
N. de Bruijn. The mathematical language automath, its usage and some of its extensions. In Symp. on automatic demonstration, pages 29–61. Springer Verlag, 1970. Lecture Notes in Mathematics Vol. 125.
J.-Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159–192, 1986.
W. Howard. The formulae-as-types notion of construction. To H.B. Curry: Essays on combinatory logic, λ-calculus and formalism, pages 479–490, 1980.
Jean-Louis Krivine. Lambda-Calcul: Types et Modèles, Etudes et Recherches en Informatique. Masson, 1990.
Jean-Louis Krivine and Michel Parigot. Programming with proofs. Inf. Process. Cybern., EIK 26(3):149–167, 1990.
Daniel Leivant. Typing and computational properties of lambda expressions. Theoretical Computer Science, 44:51–68, 1986.
Daniel Leivant. Contracting proofs to programs. Logic and Computer Science, pages 279–327, 1990.
P. Martin-Löf. Lecture notes on the domain interpretation of type theory. In Programming Methodology Group, editor, Workshop on the Semantics of Programming Languages, Göteborg, Sweden, 1983. Chalmers University of Technology.
Paul Francis Meadler. Inductive definition in type theory. PhD thesis, Cornell University, 1988.
Michel Parigot. On the representation of data in lambda-calculus. Lecture Notes in Computer Science, 440:309–321, 1989.
Michel Parigot. Recursive programming with proofs. Theoritical Computer Science, 94:335–356, 1992.
Michel Parigot and Paul Rozière. Constant time reduction in λ-calculus. Lecture Note in Computer Science, 711:608–617, 1993.
C. Paulin-Mohring. Inductive definitions in the calculus of constructions. Technical report, INRIA, 1989. Technical Report Number 110.
Christophe Raffalli. L'arithmétique fonctionnelle du second ordre avec points fixes. PhD thesis, Université Paris 7, Février 1994.
M. Tatsuta. Realizability interpretation of coinductive definitions and program synthesis using streams. In Proceedings of the fifth generation computer systems, pages 666–673. ICOT, 1992.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Raffalli, C. (1994). Data types, infinity and equality in system AF 2 . In: Börger, E., Gurevich, Y., Meinke, K. (eds) Computer Science Logic. CSL 1993. Lecture Notes in Computer Science, vol 832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0049337
Download citation
DOI: https://doi.org/10.1007/BFb0049337
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58277-9
Online ISBN: 978-3-540-48599-5
eBook Packages: Springer Book Archive