Associative-Commutative Reduction Orderings

Rewrite systems are sets of directed equations used to compute by repeatedly replacing subterms in a given expression by equal terms until a simplest form possible (a normal form) is obtained. If a rewrite system is terminating (i.e., allows no infinite sequence of rewrites), then every expression has a normal form. A variety of orderings, called reduction orderings, have been designed for proving termination, but most of them are not applicable to extended rewrite systems, where rewrites take into account inherent properties of given functions such as associativity and commutativity. In this paper we show how an ordering represented as a schematic rewrite system—the lexicographic path ordering— can be systematically modified into an ordering compatible with associativity and commutativity.