Abstract Critical Pairs and Confluence of Arbitrary Binary Relations

Abstract

In a seminal paper, Huet introduced abstract properties of term rewriting systems, and the confluence analysis of terminating term rewriting systems by critical pairs computation. In this paper, we provide an abstract notion of critical pair for arbitrary binary relations and context operators. We show how this notion applies to the confluence analysis of various transition systems, ranging from classical term rewriting systems to production rules with constraints and partial control strategies, such as the Constraint Handling Rules language CHR. Interestingly, we show in all these cases that some classical critical pairs can be disregarded. The crux of these analyses is the ability to compute critical pairs between states built with general context operators, on which a bounded, not necessarily well-founded, ordering is assumed. Dedicated to Gérard Huet on his 60th birthday. Obviously, strong confluence implies confluence, and by Newman's lemma we know that a noetherian reduction relation is confluent if and only if it is locally confluent [9]. A pair (e 1 , e 2) of elements in E is →-joinable if e 1 ↓e 2 , and →-strongly joinable if e ↓ e 2. A set of pairs will be said →-joinable if all its pairs are →-joinable.

On Combining Backward and Forward Chaining in Constraint Logic Programming

“…This suggests that the confluence proof techniques for classical CHR-such as those based on local confluence [2], strong confluence [16], or decreasing diagrams [14]-can be extended to deal with the semantics presented here.…”

(Co-)Inductive semantics for Constraint Handling Rules

“…The theorem we give next states soundness and completeness of the hybrid transition system semantics of confluent programs, provided the states respect a data-sufficiency property. In this paper we do not address the problem of proving confluence of hybrid programs, but claim it can be tackled by extending straightforwardly the work of Abdennadher et al (1999) or by adequately instanstiating the notion of abstract critical pair that we have proposed in a previous work (Haemmerlé and Fages 2007).…”

As time goes by: Constraint Handling Rules

“…We should also mention the work on an and-compositional semantics for CHR (Delzanno, Gabbrielli et al 2005;Gabbrielli and Meo 2009), which allows one to 1 Early work on CHR, as well as some more recent publications (e.g., Bouissou 2004;Duck, Stuckey et al 2007;Haemmerlé and Fages 2007), use a token store instead of a propagation history (this explains the convention of denoting the propagation history with T). A token store contains a token for every potential (future) propagation rule application, which is removed when the rule is actually applied.…”

“…Raiser and Tacchella (2007) extended the notion of confluence to non-terminating programs. Haemmerlé and Fages (2007) develop a notion of abstract critical pairs for rewriting systems in general. They illustrate this notion for CHR's theoretical semantics.…”