Termination of combined (rewrite and λ-calculus) systems

Loría-Sáenz, Carlos; Steinbach, Joachim P.

doi:10.1007/3-540-56393-8_10

Cited by 10 publications

(8 citation statements)

References 4 publications

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“…These orderings are either quite weak [Loría-Sáenz and Steinbach 1992;Jouannaud · 3 and Rubio 1998], or need a heavy user interaction [van de Pol and Schwichtenberg 1995]. Besides, they operate on terms in η-long β-normal form, hence apply only to the higher-order rewriting "à la Nipkow" [Mayr and Nipkow 1998], based on higherorder pattern matching modulo βη.…”

Section: Introductionmentioning

confidence: 99%

Polymorphic higher-order recursive path orderings

Jouannaud

Rubio

2007

J. ACM

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This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by defining a family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. These relations can be generated from two given well-founded orderings, on the function symbols and on the type constructors. The obtained orderings on terms are well-founded, monotonic, stable under substitution and include β-reductions. They can be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules using firstorder pattern matching. For example, the polymorphic version of Gödel's recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Many non-trivial examples are given which exemplify the expressive power of these orderings. All have been checked by our implementation.This paper is an extended and improved version of [Jouannaud and Rubio 1999]. Polymorphic algebras have been made more expressive than in our previous framework. The intuitive notion of a polymorphic higher-order ordering has now been made precise. The higher-order recursive path ordering itself has been made much more powerful by replacing the congruence on types used there by an ordering on types satisfying some abstract properties. Besides, using a restriction of Dershowitz's recursive path ordering for comparing types, we can integrate both orderings into a single one operating uniformly on both terms and types.

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Section: Introductionmentioning

confidence: 99%

Polymorphic higher-order recursive path orderings

Jouannaud

Rubio

2007

J. ACM

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“…[LSS92,vdP96,SWS01,JB04,FK12]) or, vice versa, adapt to rewriting techniques developed for λ-calculus (e.g. [JO91,Bla04,BR06]).…”

Section: Introductionmentioning

confidence: 99%

Termination of rewrite relations on λ-terms based on Girard's notion of reducibility

Blanqui

2016

Theoretical Computer Science

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International audienceIn this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of β-reduction in the polymorphic λ-calculus, to prove the termination of various kinds of rewrite relations on λ-terms, including rewriting modulo some equational theory and rewriting with matching modulo βη, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

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“…We have already studied the question, and the answer is affirmative. As an aplication of the present work, we have obtained such an ordering which significantly improves the few attempts that we know of [13,14,11].…”

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confidence: 55%

“…Example 6 (inspired by [13]) Let S = fListg, S 8 = f g, F = fnil : List; cons : List ! List; fcons : !…”

Section: Examplementioning

confidence: 99%

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The higher-order recursive path ordering

Jouannaud

Rubio²

Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)

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This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. The obtained ordering is well-founded, compatible with -reductions and with polymorphic typing, monotonic with respect to the function symbols, and stable under substitution. It can therefore be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules. For example, the polymorphic version of Gödel's recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Several other non-trivial examples are given which examplify the expressive power of the ordering.

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Termination of combined (rewrite and λ-calculus) systems

Cited by 10 publications

References 4 publications

Polymorphic higher-order recursive path orderings

Polymorphic higher-order recursive path orderings

Termination of rewrite relations on λ-terms based on Girard's notion of reducibility

The higher-order recursive path ordering

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