Higher-Order Termination: From Kruskal to Computability

Blanqui, Frédéric; Jouannaud, Jean-Pierre; Rubio, Albert

doi:10.1007/11916277_1

Cited by 11 publications

(6 citation statements)

References 23 publications

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“…On the other hand, we hope that the more complex closure mechanism can be improved and integrated to the ordering. These improvements are carried out in part in [Blanqui et al 2006].…”

Section: Comparisonsmentioning

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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“…On the other hand, we hope that the more complex closure mechanism can be improved and integrated to the ordering. These improvements are carried out in part in [Blanqui et al 2006].…”

Section: Comparisonsmentioning

confidence: 99%

Polymorphic higher-order recursive path orderings

Jouannaud

Rubio

2007

J. ACM

Self Cite

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“…The ordering and the computability closure definitions turn out to share many similar constructs, raising expectations for a simpler and yet more expressive definition, instead of a pair of mutually inductive definitions for the computability closure and the ordering itself, as advocated in [17]. These expectations were partly met, on the one hand in [15] with a single computability oriented definition, and on the other hand in [18] where a new, syntax oriented recursive definition was given for HORPO.…”

Section: Introductionmentioning

confidence: 99%

The Computability Path Ordering: The End of a Quest

Blanqui

Jouannaud

Rubio

2008

Computer Science Logic

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In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability argumentsà la Tait and Girard, therefore explaining the name of the improved ordering.

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“…[91,114,59]. While S-expression rewrite systems and in particular applicative systems have been studied extensively, relative little effort has been spent to prove termination of higher-order rewrite systems directly, see [88,89,67,25,90,87].…”

Section: Contributionsmentioning

confidence: 99%

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

Moser

2009

Preprint

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dedicated to Claudia Contents 9.4.1 From Cycle Analysis to Path Detection 9.4.2 Refinement Based on Path Detection 9.5 Conclusion v Preface This cumulative habilitation thesis is based on the following publications studying the complexity of term rewrite systems. It goes without saying that for all included papers I wrote the manuscript. (i) G. Moser and A. Weiermann. Relating derivation lengths with the slow-growing hierarchy directly. In Proceedings of the 14th International Conference on Rewriting Techniques and Applications, number 2706 in LNCS, pages 296-310, Springer Verlag, 2003.Own Contribution: Andreas Weiermann introduced me to the field of complexity analysis of term rewriting systems and suggested the topic of the paper. He made suggestions towards the proof methodology, but I was in charge to fill in the details.Hence I crafted the central notions and theorems and proved their correctness.(ii) T. Arai and G. Moser. Proofs of termination of rewrite systems for polytime functions. In

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Higher-Order Termination: From Kruskal to Computability

Cited by 11 publications

References 23 publications

Polymorphic higher-order recursive path orderings

Polymorphic higher-order recursive path orderings

The Computability Path Ordering: The End of a Quest

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

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