A Characterization of Weakly Church-Rosser Abstract Reduction Systems That Are Not Church-Rosser

reduction systems (ARS) are abstract structures for studying general properties of rewriting systems. An ARS is just a set equipped with a set of binary relations. Despite this generality many aspects of rewriting systems can be approached in the theory of ARS, such as confluence and normalization. Moreover, important results have been stated and proved in this setting (examples are Newman’s lemma and the Hindley–Rosen theorem [1]). Also the relationships between the Church–Rosser (CR or confluence) and weak Church–Rosser (WCR) properties can be studied in the ARS setting. Indeed, well-known counterexamples of WCR⇒ CR were abstractly described as ARS by Rosser, Newman, Hindley, Klop, and others [2, 5, 9, 7] (Fig. 1 and 2, displayed below, show some of these counterexamples). In this work, we study the structure of WCR ARS that are not CR, looking for a graph-theoretic characterization of such ARS in terms of a suitable class of reduction graphs, such that in every WCR not CR ARS, we can embed at least one element of such a class. The main result of this work is that there exists, indeed, a familyBS of WCR not CR reduction graphs such that, for every WCR not CR ARS A, there is an element of BS that can be “embedded” in A for a suitable notion of embedding. This notion of embedding is reminiscent of the well-known notion of homeomorphic subgraph (minor) in graph theory (see for example [10]). The family BS strongly differs, in its general structure, from the above-mentioned family of well-known counterexamples. It is therefore natural to ask what additional conditions can enforce a better-defined shape to WCR not CR ARS. We answer this question by singling out some natural restrictions that, once assumed, strongly specify the structure of a WCR not CR ARS. 1 Research partially supported by Murst 40% Tosca Project. 137