Completeness results for basic narrowing

Middeldorp, Aart; Hamoen, Erik

doi:10.1007/bf01190830

Cited by 85 publications

(98 citation statements)

References 14 publications

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“…Moreover, by Lemma 4, since s → p,R t , there exists a term t such that sϑ σ → p,R t in R with t = t . Since both ϑ and σ are constructor substitutions, s| p is not a variable and, thus, by definition of narrowing (see, e.g., the completeness result of [53,Lemma 3.4]), we have s ; R u with uθ = t for some constructor substitution θ .…”

Section: Now We Can Already Proceed With the Proof Of Lemmamentioning

confidence: 99%

Termination of narrowing via termination of rewriting

Nishida

Vidal

2010

AAECC

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Narrowing extends rewriting with logic capabilities by allowing logic variables in terms and by replacing matching with unification. Narrowing has been widely used in different contexts, ranging from theorem proving (e.g., protocol verification) to language design (e.g., it forms the basis of functional logic languages). Surprisingly, the termination of narrowing has been mostly overlooked. In this work, we present a novel approach for analyzing the termination of narrowing in left-linear constructor systems-a widely accepted class of systems-that allows us to reuse existing methods in the literature on termination of rewriting.

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Section: Now We Can Already Proceed With the Proof Of Lemmamentioning

confidence: 99%

Termination of narrowing via termination of rewriting

Nishida

Vidal

2010

AAECC

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“…innermost rewriting modulo AC despite of the free case [20], i.e., there are innermost rewriting sequences modulo AC that are not lifted to basic narrowing modulo Ax. And, therefore, basic narrowing modulo AC is not variant-complete.…”

Section: Basic Narrowing Modulo Is Neither Variant-complete Nor Optimmentioning

confidence: 99%

“…It is well-known that in the free case (when Ax = ∅) basic narrowing is complete for unification in the sense of lifting all innermost rewriting sequences (see [20]). That is, given a term t and a substitution σ, every innermost rewriting sequence starting from tσ can be lifted to a basic narrowing sequence from t computing a substitution more general than σ.…”

Section: Example 1 [4]mentioning

confidence: 99%

Folding Variant Narrowing and Optimal Variant Termination

Escobar

Sasse

Meseguer

2010

Rewriting Logic and Its Applications

137

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Abstract. If a set of equations E∪Ax is such that E is confluent, terminating, and coherent modulo Ax, narrowing with E modulo Ax provides a complete E ∪Ax-unification algorithm. However, except for the hopelessly inefficient case of full narrowing, nothing seems to be known about effective narrowing strategies in the general modulo case beyond the quite depressing observation that basic narrowing is incomplete modulo AC. In this work we propose an effective strategy based on the idea of the E ∪ Ax-variants of a term that we call folding variant narrowing. This strategy is complete, both for computing E ∪ Ax-unifiers and for computing a minimal complete set of variants for any input term. And it is optimally variant terminating in the sense of terminating for an input term t iff t has a finite, complete set of variants. The applications of folding variant narrowing go beyond providing a complete E ∪ Axunification algorithm: computing the E ∪Ax-variants of a term may be just as important as computing E ∪Ax-unifiers in recent applications of folding variant narrowing such as termination methods modulo axioms, and checking confluence and coherence of rules modulo axioms.

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“…We disallow rewriting rules whose lhs is a variable and assume that the lhs contains all the variables that occur in the rule. Following the terminology in [MH94], our rules are of type 1. An unconditional rewrite rule has the form l → r. A conditional (unconditional) Σ-term rewrite system R = (Σ, R), abbreviated CTRS (TRS), consists of a finite set R of conditional (unconditional) Σ-rewrite rules.…”

Section: Examples and Experimentsmentioning

confidence: 99%

Computationally Equivalent Elimination of Conditions

Şerbănuţă

Roşu

2006

Lecture Notes in Computer Science

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An automatic and easy to implement transformation of conditional term rewrite systems into computationally equivalent unconditional term rewrite systems is presented. No special support is needed from the underlying unconditional rewrite engine. Since unconditional rewriting is more amenable to parallelization, our transformation is expected to lead to efficient concurrent implementations of rewriting.

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Completeness results for basic narrowing

Cited by 85 publications

References 14 publications

Termination of narrowing via termination of rewriting

Termination of narrowing via termination of rewriting

Folding Variant Narrowing and Optimal Variant Termination

Computationally Equivalent Elimination of Conditions

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