Solving Goals in Equational Languages

Solving equations in equational Horn-clause theories is a programming paradigm that combines logic programming and functional programming in a clean manner. Languages like EQLOG, SLOG and RITE, express programs as conditional rewrite rules and goals as equations to be solved. Procedures for completion of conditional equational theories, in a manner akin to that of Knuth and Bendix for unconditional theories, also require methods for solving equations appearing in conditions. Rewrite-based logic-programming uses (conditional) narrowing to solve equational goals. Recently a different, topdown equation solving procedure was proposed for unconditional rewrite systems. In this paper, we express equational goal solving using conditional rules. Some refinements are described: the notion of operator derivability is used to prune useless paths in the search tree and our use of oriented goals eliminates some redundant paths leading to non-normalized solutions. Our goal-directed method can also be extended to handle conditional systems. I. Equational Programming Several proposed programming languages use conditional equations as a means of combining the main features of logic programming and functional programming; such languages include RITE [Dershowitz-Plaisted-85], SLOG [Fribourg-85], and EQLOG [Goguen-Meseguer-86]. In this paradigm, a program is a set of rules, that is, directed (conditional) equations, and a goal is the question whether an equation s = t has a solution in the equational theory presented by the program. Computing consists of finding values (substitutions) for the variables in s and t for which the equality holds. Efficient methods of solving equations are therefore very important. So, too, is the ability to detect that equations are unsatisfiable. * This research was supported in part by the National Science Foundation under Grant DCR 85-13417.