Logicality of conditional rewrite systems

Yamada, Toshiyuki; Avenhaus, Jürgen; Loría-Sáenz, Carlos; Middeldorp, Aart

doi:10.1007/bfb0030592

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…A substitution θ is a solution of a goal G if Gθ → * R . Since confluence is insufficient [19] to guarantee that joinability coincides with the equality relation induced by the underlying conditional equational theory, we decided to drop confluence and use the above notion of solution. We say that θ is X-normalized for a subset X of V if every variable in X is mapped to a normal form with respect to R.…”

Section: Preliminariesmentioning

confidence: 99%

New completeness results for lazy conditional narrowing

Marin

Middeldorp

2004

Proceedings of the 6th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming

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We show the completeness of the lazy conditional narrowing calculus (LCNC) with leftmost selection for the class of deterministic conditional rewrite systems (CTRSs). Deterministic CTRSs permit extra variables in the right-hand sides and conditions of their rewrite rules. From the completeness proof we obtain several insights to make the calculus more deterministic. Furthermore, and similar to the refinements developed for the unconditional case, we succeeded in removing all nondeterminism due to the choice of the inference rule of LCNC by imposing further syntactic conditions on the participating CTRSs and restricting the set of solutions for which completeness needs to be established.

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Section: Preliminariesmentioning

confidence: 99%

New completeness results for lazy conditional narrowing

Marin

Middeldorp

2004

Proceedings of the 6th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming

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“…All this makes the trivial proof of the Church-Rosser Theorem in the unconditional case non-trivial for conditional theories. Indeed, even for CTRSs (i.e., Σ unsorted and B = ∅), it is well-known that the Church-Rosser theorem does not hold for an arbitrary confluent theory R without imposing additional conditions on R (see [72] and Example 11 below).…”

Section: Rosser Theorem Modulo B Showing That Under Suitable Conditmentioning

confidence: 99%

“…The Church-Rosser property for CTRSs (called there logicality) has been carefully studied in [72], where several sufficient conditions on R ensuring the Church-Rosser Theorem are given. Let me briefly summarize a class of CTRSs enjoying the Church-Rosser property proposed (among other classes) in [72]. Given a CTRS (Σ, R), call a Σ-term t R-irreducible iff there is no t such that t → R t .…”

Section: The Church-rosser Property For Conditional Rewriting Modulo Bmentioning

confidence: 99%

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Strict coherence of conditional rewriting modulo axioms

Meseguer

2017

Theoretical Computer Science

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Conditional rewriting modulo axioms with rich types makes specifications and declarative programs very expressive and succinct and is used in all well-known rule-based languages. However, the current foundations of rewriting modulo axioms have focused for the most part on the unconditional and untyped case. The main purpose of this work is to generalize the foundations of rewriting modulo axioms to the conditional order-sorted case. A related goal is to simplify such foundations. In particular, even in the unconditional case, the notion of strict coherence proposed here makes rewriting modulo axioms simpler and easier to understand. Properties of strictly coherent conditional theories, like operational equi-termination of the → R/B and → R,B relations and general conditions for the conditional Church-Rosser property modulo B are also studied.

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“…Ce résultat est donc une généralisation du résultat de logicalité démontré par G. Birkhoff dans le cadre de la logique équationnelle mono-sorte et de toutes ses extensions algébriques simples ( [30], [31]). Il établit un résultat de complétude entre la preuve formelle et la réécriture utilisée comme un mécanisme de déduction (les aspects opérationnels sont ignorés).…”

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Une approche générique de la réécriture

Aiguier¹,

Bahrami²

2003

TSI

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Dans cet article, nous étudions la réécriture en tant que système de preuve opérationnel de façon générique (c'est-à-dire indépendamment de la logique sous-jacente). Dans ce but, nous proposons des conditions simples qui permettent de caractériser les logiques où ce type de preuve effective peut être appliqué. Ces conditions sont fondées sur une propriété définie sur les arbres de preuve, que nous appelons semi-commutation. Nous montrons alors comment axiomatiser les notions usuelles de terminaison, confluence et confluence locale au sein de ce méta-formalisme. Cela nous permet de retrouver les résultats fondamentaux de Church-Rosser et Newman et de proposer une méthode de complétion de Knuth et Bendix générique. ABSTRACT. In this paper we study rewriting as an operational proof system in a generic (i.e. logic-independent) way. To do this, we propose simple conditions which allow to characterise logics were this kind of effective proof can be applied. This conditions are based on a proof tree property, that we call semi-commutation. We then show how to define the standard notion of termination, confluence and local confluencec in this meta-formalism. It allows us to obtain the fundamental results of Church-Rosser and Newman and to propose a generic Knuth and Bendix completion method. MOTS-CLÉS : réécriture abstraite, logicalité, propriété de Church-Rosser, lemme de Newman, système de Knuth et Bendix, complétion de Knuth et Bendix KEY WORDS : abstract rewriting, logicality, Church-Rosser property, Diamond lemma, Knuth&Bendix system, Knuth&Bendix completion Technique et sciences informatiques.

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Logicality of conditional rewrite systems

Cited by 3 publications

References 8 publications

New completeness results for lazy conditional narrowing

New completeness results for lazy conditional narrowing

Strict coherence of conditional rewriting modulo axioms

Une approche générique de la réécriture

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