Abstract
The essentially free variables of a term t in some λ-calculus, FV β (t), form the set {x ¦ ∀u.t = β u ⇒ x ∈ FV(u)}. This set is significant once we consider equivalence classes of λ-terms rather than λ-terms themselves, as for instance in higher-order rewriting. An important problem for (generalised) higher-order rewrite systems is the variable containment problem: given two terms t and u, do we have for all substitutions θ and contexts C[]that FV β (C[t] θ)\(\supseteq\)FV β (C[u θ])?
This property is important when we want to consider t → u as a rewrite rule and keep n-step rewriting decidable. Variable containment is in general not implied by FV β (t)\(\supseteq\)FV β (u). We give a decision procedure for the variable containment problem of the second-order fragment of λ→. For full λ→ we show the equivalence of variable containment to an open problem in the theory of PCF; this equivalence also shows that the problem is decidable in the third-order case.
The research reported here was partially supported by SERC grant GR/J07303.
Preview
Unable to display preview. Download preview PDF.
References
Hendrik P. Barendregt. The Lambda-Calculus, its Syntax and Semantics. North-Holland, 1984.
Hendrik P. Barendregt. Introduction to generalised type systems. Journal of Functional Programming, 1(2):124–154, 1991.
Hendrik P. Barendregt. Lambda calculi with types. In Handbook of Logic in Computer Science, Vol.2, pages 117–309. Oxford Science Publications, 1992.
Gilles Dowek. Third order matching is decidable. In Proceedings of the 7th Symposium on Logic in Computer Science, pages 2–10, 1992.
Philippa Gardner. Representing Logics in Type Theory. PhD thesis, University of Edinburgh, 1992.
W. D. Goldfarb. The undecidability of the second-order unification problem. Theoretical Computer Science, 13:225–230, 1981.
Gérard Huet and Bernard Lang. Proving and applying program transformations expressed with second-order patterns. Acta Informatica, 11:31–55, 1978.
Achim Jung and Allen Stoughton. Studying the fully abstract model of PCF within its continuous function model. In Typed Lambda Calculi and Applications, 1993. LNCS 664.
Stefan Kahrs. Towards a domain theory for termination proofs. In Rewriting Techniques and Applications, pages 241–255, 1995. LNCS 914.
Jan Willem Klop. Combinatory Reduction Systems. PhD thesis, Centrum voor Wiskunde en Informatica, 1980.
Jan Willem Klop, Vincent van Oostrom, and Femke van Raamsdonk. Combinatory reduction systems: Introduction and survey. Theoretical Computer Science, 121:279–308, 1993.
Ralph Loader. The undecidability of λ-definability, 1994.
Saunders MacLane. Categories for the Working Mathematician. Springer, 1971.
Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Technical Report TUM-I94333, Technische Universität München, 1994.
Aart Middeldorp. Modular aspects of properties of term rewriting systems related to normal forms. In Rewriting Techniques and Applications, pages 263–277, 1989. LNCS 355.
Tobias Nipkow. Higher order critical pairs. In Proceedings of the 6th Symposium on Logic in Computer Science, pages 342–349, 1991.
Vincent Padovani. On equivalence classes of interpolation equations. In Typed Lambda Calculi and Applications, pages 335–249, 1995. LNCS 902.
Jaco van de Pol. Termination proofs for higher-order rewrite systems. In Higher-Order Algebra, Logic, and Term Rewriting, pages 305–325, 1993. LNCS 816.
Jaco van de Pol and Helmut Schwichtenberg. Strict functionals for termination proofs. In Typed Lambda Calculi and Applications, pages 350–364, 1995. LNCS 902.
Kurt Sieber. Reasoning about sequential functions via logical relations. In M.P. Fourman, P.T. Johnstone, and A.M. Pitts, editors, Applications of Categories in Computer Science, pages 258–269. Cambridge University Press, 1992.
Marek Zaionc. Lambda definability is decidable for second order types and for regular third order types. Unpublished Manuscript, University of New York at Buffalo, 1995.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kahrs, S. (1996). The variable containment problem. In: Dowek, G., Heering, J., Meinke, K., Möller, B. (eds) Higher-Order Algebra, Logic, and Term Rewriting. HOA 1995. Lecture Notes in Computer Science, vol 1074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61254-8_22
Download citation
DOI: https://doi.org/10.1007/3-540-61254-8_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61254-4
Online ISBN: 978-3-540-68389-6
eBook Packages: Springer Book Archive