A critical pair criterion for completion modulo a congruence

Rewrite systems axe collections of directed equations (rules) used to compute by repeatedly replacing subterms in a given formula until a simplest form possible (normal form) is obtained. Many formula manipulation systems such as REDUCE or MAGSYMA use equations for simplification in this manner. Canonical (i.e., terminating Church-Rosser) rewrite systems have the property that all equal terms (and only equal terms) simplify to an identical normal form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient. A number of canonical systems have been derived with the Knuth-Bendix completion procedure [5]. Unfortunately, the Knuth-Bendix procedure can not be applied to axioms such as commutativity that induce non-terminating rewrite sequences. There are also some practical tilrdtations in its handling of associativity, as pointed out by Peterson and Stickel [6]. Associativity and commutafivity are typical equations that are more naturally viewed as "structural" axioms (defining a congruence relation on terms) rather than as "simplifiers" (defining a reduction relation). Given a set of axioms A and a rewrite system R, we denote by RA the corresponding relation of rewri t ing m o d u t o A , defned ~as the application of rules in R via A-matching. For example, if A consists of the associativity and commutafivity axioms for addition, and the rewrite system R consists of a single rule f (x ,z ) --~ x, then f ( x q y , y ÷ ~ ) can not be rewritten by R, whereas it can be rewritten to x + y (and y qz) by RA. Extensions of the Knuth-Bendix procedure to rewriting modulo a congruence have been described by Peterson and Stickel [6] (for sets A of associativity and commutativity axioms), by Jouannaud and Kirchner [3], and Bachmair and Dershowitz [1]. The fundamental operations in these procedures are A-matching and A-uniflcation. Two terms s and t are said to be A-unit~abIe if and only if there is a substitution ~r (called an A-urd/]er), such that s~r and t~r are equivalent with respect to A. The above-mentioned completion procedures apply to theories for which complete sets of A-mfifiers can be computed. If u ~ v and s --~ t are rules and p is a non-vaxiable position in u, such that u / p and s are A-unifiable with a complete set of unifiers E, then the "rewriting ambiguity" va e-R u~ "-~n~ u(r[ta] determines an A-crit ical pair vcr = ug[t~r], for each ~ in E. Completion augments a giver~ rewrite system R by so-cMled "extended" rules 2 and then systematically computes A-critical pairs to check whether the two terms wr aud ue[tcr] reduce to an identical normal form. If the test is successful for all criticM pairs, then the rewrite system is canonical (in the sense of defining normal forms that are unique up to equivalence in A). Otherwise, the offending equations have to be turned into rules and critical pair computation continues with the new rules. (It is possible that endless new rules are generated and completion does not terminate. Completion may even fail, when an equation can not be oriented into a rule.) The most expensive part of completion is the reduction of terms to normal form. Critical pairs in which both terms reduce to identical normal forms are redundant. Various techniques~ called critical pair criteria, have been proposed for standard completion for detecting redundancies more efficiently than by normalization of terms (see Baehmair and Dershowitz [2] for an overview). We sketch a similar technique for rewriting modulo a congruence.