A Dependency Pair Framework for AvC-Termination

The development of powerful techniques for proving termination of rewriting modulo a set of equational axioms is essential when dealing with rewriting logic-based programming languages like CafeOBJ, Maude, ELAN, OBJ, etc. One of the most important techniques for proving termination over a wide range of variants of rewriting (strategies) is the dependency pair approach. Several works have tried to adapt it to rewriting modulo associative and commutative (AC) equational theories, and even to more general theories. However, as we discuss in this paper, no appropriate notion of minimality (and minimal chain of dependency pairs) which is well-suited to develop a dependency pair framework has been proposed to date. In this paper we carefully analyze the structure of in nite rewrite sequences for rewrite theories whose equational part is any combination of associativity and/or commutativity axioms, which we call AvC-rewrite theories. Our analysis leads to a more accurate and optimized notion of dependency pairs through the new notion of stably minimal term. We then develop a suitable dependency pair framework for proving termination of AvC-rewrite theories.Alarcón Jiménez, B.; Gutiérrez Gil, R.; Lucas, S.; Meseguer, J. (2011). A Dependency Pair Framework for AvC-Termination. http://hdl.handle.net/10251/1079

AC Dependency Pairs Revisited

Rewriting modulo AC, i.e., associativity and/or commutativity of certain symbols, is among the most frequently used extensions of term rewriting by equational theories. In this paper we present a generalization of the dependency pair framework for termination analysis to rewriting modulo AC. It subsumes existing variants of AC dependency pairs, admits standard dependency graph analyses, and in particular enjoys the minimality property in the standard sense. As a direct benefit, important termination techniques are easily extended; we describe usable rules and the subterm criterion for AC termination, which properly generalize the non-AC versions. We also perform these extensions within IsaFoR - the Isabelle formalization of rewriting - and thereby provide the first formalization of AC dependency pairs. Consequently, our certifier CeTA now supports checking proofs of AC termination