Induction for termination with local strategies - Extended version -

Rapport interne.In this paper, we propose a method for specifically proving termination of rewriting with particular strategies: local strategies on operators. An inductive proof procedure is proposed, based on an explicit induction on the termination property. Given a term, the proof principle relies on alternatively applying the induction hypothesis on its subterms, by abstracting the subterms with induction variables, and narrowing the obtained terms in one step, according to the strategy. The induction relation, an F-stable ordering having the subterm property, is not given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints

Proving weak termination also provides the right way to terminate - extended version -

Rapport interne.From an inductive method for proving weak innermost termination of rule-based programs, we automatically infer, for each successful proof, a finite strategy for data evaluation. The proof principle uses an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we analyse proof trees built from the rewrite system, schematizing the innermost derivations of any ground term. These proof trees are issued from patterns representing sets of ground terms. They are generated using two mechanisms, namely abstraction, introducing variables that represent ground terms in normal form, and narrowing, schematizing rewriting on ground terms. From the proof trees, we can extract for any given ground term, a rewriting strategy to compute one of its normal form, even if the ground term admits infinite rewriting derivations

CARIBOO: An Induction Based Proof Tool for Termination with Strategies

Colloque avec actes et comité de lecture. internationale.International audienceWe describe Cariboo, the implementation of an inductive process recently proposed to prove termination of rewriting under strategies on ground term algebras. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. It applies in particular to non-terminating systems which are terminating with innermost or local strategies. The narrowing process, well known to easily diverge, is controlled by using appropriate abstraction constraints. The abstraction mechanism lies on satisfiability of ordering constraints. Thanks to the power of induction, these ordering constraints are often simple and automatically solved by our system. Otherwise, they can be treated by the user or any external automatic solver. On many examples, Cariboo even enables to succeed without considering any constraint at all; the process is then completely automatic. Cariboo offers a graphical view of the proof process. It is implemented in ELAN, a rule based programming environment, and so can be used for proving termination of ELAN programs

CARIBOO: An Induction Based Proof Tool for Termination with Strategies -- Extended version--

Version étendue d'un article publié dans les actes de :"Fourth International Conference on Principles and Practice of Declarative Programming", Pitsburgh, USA, Octobre 2002, ACM Press. Rapport interne.We describe Cariboo, the implementation of an inductive process recently proposed to prove termination of rewriting under strategies on ground term algebras. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. It applies in particular to non-terminating systems which are terminating with innermost or local strategies. The narrowing process, well known to easily diverge, is controlled by using appropriate abstraction constraints. The proof mechanism lies on abstraction and ordering constraints satisfiability problems. Thanks to the power of induction, ordering constraints are often simple and automatically solved by our system. Otherwise, they can be treated by the user or any external automatic solver. On many examples, Cariboo even enables to succeed without considering any constraint at all; the process is then completely automatic. Cariboo offers a graphical view of the proof process. It is implemented in ELAN, a rule based programming environment, and so can be used for proving termination of ELAN programs

Termination of ELAN strategies by simplification - Extended version -

Rapport interne.We propose a transformation based method for proving termination of ELAN strategies. We first give a sufficient criterion for ELAN strategies to terminate, only lying on rewrite rules involved in the strategy. We then give a simplification process of strategies, itself described by rewriting, to empower the previous criterion. This simplification, beyond easing termination proof of strategies, can both facilitate elaboration of specifications and ease proofs of other program properties

Extensional and Intensional Strategies

This paper is a contribution to the theoretical foundations of strategies. We first present a general definition of abstract strategies which is extensional in the sense that a strategy is defined explicitly as a set of derivations of an abstract reduction system. We then move to a more intensional definition supporting the abstract view but more operational in the sense that it describes a means for determining such a set. We characterize the class of extensional strategies that can be defined intensionally. We also give some hints towards a logical characterization of intensional strategies and propose a few challenging perspectives

Terminaison de la réécriture sous stratégies

Texte intégral accessible uniquement aux membres de l'Université de LorraineThe aim of this thesis is to study and develop tools for proving termination of rule-based languages using strategies. Starting from an original method for proving, by induction, the termination of innermost rewriting, we enhanced and extended this method to the outermost and local strategies. These proof processes have then been implemented in a tool, named CARIBOO. Such languages as ELAN make it possible for the programmer to define his own strategies, by combining rules of the program with appropriate strategy operators. We came up with a method allowing to prove termination of such strategies, based on an automatic simplification process. Finally, our original inductive proof process has been adapted to show weak termination of programs. This new process provides both the proof of termination of at least one evaluation of any data and an evaluation algorithm for this data.L'objectif de cette thèse est l'étude et la réalisation d'outils pour prouver la terminaison de programmes à base de règles utilisant des stratégies. Partant d'une méthode originale permettant de prouver par induction la terminaison de la réécriture innermost, nous avons amélioré et étendu ce processus de preuve à la stratégie outermost puis aux stratégies locales. Ces processus de preuve ont été implantés dans un outil nommé CARIBOO. Des langages tels qu'ELAN permettent à l'utilisateur de définir ses propres stratégies, par combinaison des règles du programme au moyen d'opérateurs adaptés. Nous avons élaboré une méthode de preuve de terminaison de ces stratégies, basée sur un processus automatique de simplification. Enfin, nous avons adapté notre processus de preuve inductif original à la preuve de la terminaison faible d'un programme, qui fournit à la fois la preuve de l'existence d'une évaluation terminante de toute donnée et un algorithme d'évaluation de cette donnée

Terminaison de la réécriture sous stratégies

L'objectif de cette thèse est l'étude et la réalisation d'outils pour prouver la terminaison de programmes à base de règles utilisant des stratégies. Partant d'une méthode originale permettant de prouver par induction la terminaison de la réécriture innermost, nous avons amélioré et étendu ce processus de preuve à la stratégie outermost puis aux stratégies locales. Ces processus de preuve ont été implantés dans un outil nommé CARIBOO. Des langages tels qu'ELAN permettent à l'utilisateur de définir ses propres stratégies, par combinaison des règles du programme au moyen d'opérateurs adaptés. Nous avons élaboré une méthode de preuve de terminaison de ces stratégies, basée sur un processus automatique de simplification. Enfin, nous avons adapté notre processus de preuve inductif original à la preuve de la terminaison faible d'un programme, qui fournit à la fois la preuve de l'existence d'une évaluation terminante de toute donnée et un algorithme d'évaluation de cette donnée.The aim of this thesis is to study and develop tools for proving termination of rule-based languages using strategies. Starting from an original method for proving, by induction, the termination of innermost rewriting, we enhanced and extended this method to the outermost and local strategies. These proof processes have then been implemented in a tool, named CARIBOO. Such languages as ELAN make it possible for the programmer to define his own strategies, by combining rules of the program with appropriate strategy operators. We came up with a method allowing to prove termination of such strategies, based on an automatic simplification process. Finally, our original inductive proof process has been adapted to show weak termination of programs. This new process provides both the proof of termination of at least one evaluation of any data and an evaluation algorithm for this data.NANCY1-SCD Sciences & Techniques (545782101) / SudocSudocFranceF

Induction for innermost and outermost ground termination

Rapport interne.We propose an original approach to prove termination of innermost rewriting on ground term algebras, based on induction, abstraction and narrowing. Our method applies in particular to non-terminating systems which are innermost terminating, and to systems that do not innermost terminate on the free term algebra but do on the ground term one. The induction relation, an F-stable ordering having the subterm property, is not given a priori, but its existence is checked along the proof, by testing satisfiability of ordering constraints. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. An extension of the method is given, where the noetherian induction is strengthened by a structural induction. A variant is also proposed, to characterize terminating subset of the ground term algebra, for non-innermost terminating system. Finally, the method is adapted in a natural way to outermost termination