CiME: Completion modulo E

Abstract

Completion is an algorithm for building convergent rewrite systems from a given equational axiomatization. The story began in 1970 with the well-known KnuthBendix completion algorithm [8]. Unfortunately, this algorithm was not able to deal with simple axioms like commutativity (z + y = y + x) because such equations cannot be oriented into a terminating rewrite system. This problem have been solved by the so-called AC-completion algorithm of Lankford and Ballantyne [9] and Peterson and Stickel [14], which is able to deal with any permutative axioms, the most popular being assoeiativity and commutativity. In 1986, Jouannand and Kirchner [6] introduced a general T-completion algorithm which was able to deal with any theory T provided that T-congruence classes are finite, and in 1989, Bachmair and Dershowitz extended it to the case of any T such that t h e subterm relation modulo T is terminating. Because of these restrictions, these algorithms are not able to deal with the most interesting cases, AC plus unit (z + 0 = x denoted ACU) being the main one. The particular case of ACU has been investigated first in 1989 by Peterson, Baird and Wilkerson [1]: they used constrained rewriting to avoid the non-termination problem; and an ACU-completion algorithm has been described then by Jouannand and March~ in 1990 [7]. Independently from this story, in the domain of computer algebra, an algorithm for computing Gr6bner bases of polynomial ideals has been found by Buchberger in 1965 [3] and much later than that, in 1981, Loos and Buchberger [11, 4] remarked that this algorithm and the previous completion algorithms behave in a very similar way. The problem of unifying these two algorithms into a common general one arised. In 1993, using the ideas introduced for ACU-completion, Marchd described a new completion algorithm based on a variant of rewriting modulo T: normalized rewriting [12, 13], where terms have to be normalized with respect to a convergent rewrite system S equivalent to T. Of course, this assumes the existence of such an S, but this appears to be true for the examples we were interested in: AC plus unit, AC plus idempotence (x + x = x), nilpotence (x + x = 0), Abelian group theory, commutative ring theory, Boolean ring theory, finite fields theory.