Skip to main content Accessibility help
×
Home

An insertion operator preserving infinite reduction sequences

  • DAVID CHEMOUIL (a1)

Abstract

A common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of ‘commutation’ for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form.

Hence the purpose of this paper is threefold:

  • First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified.
  • Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems.
  • Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with:
    1. (i)copies of inductive types;
    2. (ii)the representation of symmetric groups.

Copyright

References

Hide All
Baader, F. and Nipkow, T. (1998) Term Rewriting and All That, Cambridge University Press.
Bachmair, L. and Dershowitz, N. (1986) Commutation, transformation, and termination. In: Siekmann, J.H. (ed.) Proceedings of the Eighth International Conference on Automated Deduction (Oxford, England). Springer-Verlag Lecture Notes in Computer Science 230 5–20.
Barendregt, H.P. (1984) The Lambda Calculus – Its Syntax and Semantics, North-Holland.
Chemouil, D. (2005) Isomorphisms of simple inductive types through extensional rewriting. Mathematical Structures in Computer Science 15 (5)875915.
Chemouil, D. and Soloviev, S. (2003) Remarks on isomorphisms of simple inductive types. In: Geuvers, H. and Kamareddine, F. (eds.) Electronic Notes in Theoretical Computer Science 85.
Di Cosmo, R. (1995) Isomorphisms of Types: From λ-Calculus to Information Retrieval and Language Design, Progress in Theoretical Computer Science, Birkhäuser.
Di Cosmo, R. (1996a) A brief history of rewriting with extensionality. In: Kamareddine, F. (ed.) International Summer School on Type Theory and Rewriting (slides).
Di Cosmo, R. (1996b) On the power of simple diagrams. In: Ganzinger, H. (ed.) Proceedings of the 7th International Conference on Rewriting Techniques and Applications (RTA-96). Springer-Verlag Lecture Notes in Computer Science 1103 200–214.
Di Cosmo, R. and Kesner, D. (1993) Simulating expansions without expansions. Technical Report RR-1911, INRIA.
Di Cosmo, R. and Kesner, D. (1996a) Combining algebraic rewriting, extensional lambda calculi, and fixpoints. Theoretical Computer Science 169 (2)201220.
Di Cosmo, R. and Kesner, D. (1996b) Rewriting with extensional polymorphic lambda-calculus. Springer-Verlag Lecture Notes in Computer Science 1092.
Doornbos, H. and von Karger, B. (1998) On the union of well-founded relations. Logic Journal of the IGPL 6 (2)195201.
Flegontov, A. and Soloviev, S. (2002) Type theory in differential equations. In: VIII International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACA'02), Moscow.
Geser, A. (1990) Relative Termination, Ph.D. thesis, Universität Passau, Germany.
Girard, J.-Y., Lafont, Y. and Taylor, P. (1988) Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press.
Lengrand, S. (2005) Induction principles as the foundation of the theory of normalisation: Concepts and techniques. Technical report, PPS laboratory, Université Paris 7. (Available at http://hal.ccsd.cnrs.fr/ccsd-00004358.)
Matthes, R. (2000) Lambda calculus: A case for inductive definitions. Lecture notes for ESSLLI'2000 (European Summer School in Logic, Language and Information).
Soloviev, S. and Chemouil, D. (2004) Some algebraic structures in lambda-calculus with inductive types. In: Berardi, S., Coppo, M. and Damiani, F. (eds.) Proc. Types'03. Springer-Verlag Lecture Notes in Computer Science 3085.
Terese, (2003) Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science 55, Cambridge University Press.

An insertion operator preserving infinite reduction sequences

  • DAVID CHEMOUIL (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed