Goal-Directed Equation Solving

Solving equations in equational Horn-clause theories is a programming paradigm that combines logic programming and functional programming in a clean manner. Languages like E&LOG, SLOG and RITE, express programs as rewrite rules and use narrowing to solve goals expressed as equations. In this paper, we express equational goal solving by means of a logic program that simulates rewriting of terms. Our goal-directed equation solving procedure is based on “directed goals”, and combines narrowing with a more top-down approach. We also show how to incorporate a notion of operator derivability to prune some useless paths, while maintaining completeness of the method. 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 and Plaisted, 19851, SLOG [Fribourg, 19851, and EQLOG [Goguen and Meseguer, 19861. Computing consists of finding values (substitutions) for the variables in a goal s=t for which the equality holds. Efficient methods of solving equations are therefore very important, as is the ability to detect when an equation is unsatisfiable. In this paper, we concentrate on programs composed of unconditional rules though the ideas extend to conditional rules, as employed in the above languages. Solving equational goals and detecting unsatisfiability are also important for theorem-proving procedures (e.g. . [Kaplan-881) based on conditional equations. Consider the following example of a system for reversing a list used in [Ullman and Van Gelder, 19851 to illustrate their scheme of top-down capture rules, where rev is reverse and tcons adds an element to the end of a list. (We use capital letters for variables in rules and terms.) * This research was supported in Foundation under Grant DCR 85-13417. 166 Automated Reasoning part by the National rev(ni1) 4 nil rev(A l X) -+ tcons(rev(X),A) tcons (nil&) 3 A *na’! tcon8 (.&x,A j --+ B.tcons (X,A) 1 A goal of the form X=Pev(1*2.&) can be solved by rewriting the right-hand side of the goal to yield X=2al*nil; rewriting corresponds to the functional part of equational programming. On the other hand, a query like rev (X)=1-2 *nil, requires equation solving t.o produce the value(s) for X that satisfies the equation. This query has the answer, {Xt-+2*1*nil}. Finding solutions corresponds to the logic programming capability. Solving equations is, therefore, a basic operation in interpreters for such equational languages and efficient methods are of critical importance. in general, paramodulation can be used (as in resolution-based theorem provers to solve equations, but is highly inefficient. For equational theories that can be presented as a (ground) confluent rewrite system, better equation-solving methods have been devised, narrowing [Slagle-74, Fay79, Hullot-801 being the most popular. Techniques for helping make a narrowing procedure efficient are given in [Josephson-Dershowitz-861. An alternative approach to equation solving, based on decomposition and restructuring has been suggested in [Martelli,etal.-861. We will refer to the latter as the decomposition procedure. In this paper, we combine the above two approaches in a goal-directed procedure. When an equation is unsatisfiable, none of these procedures are guaranteed to halt. Indeed, this is inherent to the semi-decidabiiity of the (equational) satisfiability problem. Still, the ability to detect some unsatisfiable subgoals can save a lot of unnecessary computation. We describe the narrowing and decomposition procedures in Section 2. Our new, goal-directed procedure is formulated as a Prolog program in Section 3; it captures the advantages of both narrowing and decomposition and incorporates pruning of certain unsatisfiable goals. From: AAAI-88 Proceedings. Copyright ©1988, AAAI (www.aaai.org). All rights reserved.