Strict coherence of conditional rewriting modulo axioms

Abstract

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.

Constrained narrowing for conditional equational theories modulo axioms

“…As shown in [61], a considerably better correspondence between → R/B and → R,B , called strict coherence, can be achieved for a conditional rewrite theory R = (Σ, B, R) if the axioms B are regular and linear and the conditional rules R are closed under so-called B-extensions, a notion going back to [67]. In this section we summarize some of the key notions and results from [61].…”

“…We also present basic notions and results from [61] on the strict coherence property for conditional rewrite rules that allows the rewrite relation → R/B to be (bi-)simulated by the much simpler relation → R,B .…”

“…Given a rewrite theory R = (Σ, B, R), we follow closely the treatment in [61] to define the rewriting modulo B relation → R/B , and the easier to implement relation → R,B , by appropriate inference systems. The inference system 3 defining both → R/B and → R/B when Σ has non-empty sorts is given as follows.…”

“…A semi-algorithm to close a rewrite theory R = (Σ, B, R) under B-extensions under the above assumptions on B by computing for each rewrite rule l → r if C in R its extension closure Ext B (l → r if C) is described in [61]. The main results about the strict coherence of rewrite theories closed under B-extensions can be summarized as follows:…”

Proving Ground Confluence of Equational Specifications Modulo Axioms

“…In this paper we will mostly focus on rewrite theories of the form R E = (Σ, B, − → E ) associated to an equational theory E = (Σ, E B), were: (i) B are decidable structural axioms whose equations u = v ∈ B are linear (no repeated variables in either u or v) and regular (same variables in u and v), for which a matching algorithm exists, and (ii) the possibly conditional rewrite rules − → E are strictly B-coherent [13]. Under such assumptions, the rewrite relation t → R E u holds iff there exists u such that u = B u, and t →− → E ,B u , where, by definition, t →− → E ,B u iff there exists a rule (l → r if φ) ∈ − → E , a position p in t and a substitution θ such that t| p = B lθ, u = t[rθ] p , and R E φθ.…”