Abstract
Closed reductions in the λ-calculus is a strategy for a calculus of explicit substitutions which overcomes many of the usual syntactical problems of substitution. This is achieved by only moving closed substitutions through certain constructs, which gives a weak form of reduction, but is rich enough to capture the usual strategies in the λ-calculus (call-by-value, call-by-need, etc.) and is adequate for the evaluation of programs. An interesting point is that the calculus permits substitutions to move through abstractions, and reductions are allowed under abstractions, if certain conditions hold. The calculus naturally provides an efficient notion of reduction (with a high degree of sharing), which can easily be implemented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lévy. Explicit substitutions. Journal of Functional Programming, 1(4):375–416, 1991.
Z. M. Ariola, M. Felleisen, J. Maraist, M. Odersky, and P. Wadler. A call-by-need lambda calculus. In Proceedings of the 22nd ACM Symposium on Principles of Programming Languages (POPL’95), pages 233–246. ACM Press, 1995.
P.-L. Curien, T. Hardin, and J.-J. Lévy. Confluence properties of weak and strong calculi of explicit substitutions. Journal of the ACM, 43(2):362–397, 1996.
J.-Y. Girard. Geometry of interaction 1: Interpretation of System F. In R. Ferro, C. Bonotto, S. Valentini, and A. Zanardo, eds, Logic Colloquium 88, Studies in Logic and the Foundations of Mathematics. North Holland Publishing Company, 1989.
T. Hardin, L. Maranget, and B. Pagano. Functional back-ends within the lambdasigma calculus. Journal of Functional Programming, 8(2):131–172, 1998.
J. Kennaway and M. Sleep. Director strings as combinators. ACM Transactions on Programming Languages and Systems, 10:602–626, 1988.
P. Lescanne and J. Rouyer-Degli. The calculus of explicit substitutions lambdaupsilon. Technical Report RR-2222, INRIA, 1995.
I. Mackie. YALE: Yet another lambda evaluator based on interaction nets. In Proceedings of the 3rd ACM SIGPLAN International Conference on Functional Programming (ICFP’98), pages 117–128. ACM Press, 1998.
P.-A. Melliès. Typed lambda-calculi with explicit substitutions may not terminate. In Proceedings of the Second International Conference on Typed Lambda Calculi and Applications, number 902 in Lecture Notes in Computer Science, pages 328–334. Springer-Verlag, 1995.
C. Muñoz. Confluence and preservation of strong normalisation in an explicit substitutions calculus (extended abstract). In Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey. IEEE Computer Society Press, 1996.
G. Plotkin. LCF considered as a programming language. Theoretical Computer Science, 5(3):223–256, 1977.
K. H. Rose. Explicit substitution-tutorial and survey. Lecture Series LS-96-3, BRICS, Dept. of Computer Science, University of Aarhus, Denmark, 1996.
N. Yoshida. Optimal reduction in weak lambda-calculus with shared environments. Journal of Computer Software, 11(6):3–18, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fernández, M., Mackie, I. (1999). Closed Reductions in the λ-Calculus. In: Flum, J., Rodriguez-Artalejo, M. (eds) Computer Science Logic. CSL 1999. Lecture Notes in Computer Science, vol 1683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48168-0_16
Download citation
DOI: https://doi.org/10.1007/3-540-48168-0_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66536-6
Online ISBN: 978-3-540-48168-3
eBook Packages: Springer Book Archive