Synthesis by Completion

The Knuth-Bendix completion procedure was introduced as a means of deriving canonical term-rewriting systems to serve as deci­ sion procedures for given equational theories. The procedure gen­ erates new rewrite rules to resolve ambiguities resulting from exist­ ing rules that overlap. We propose using this procedure to syn­ thesize logic programs, as well as functional programs, from specifications and domain knowledge expressed as equivalencepreserving rewrite rules. An implementation is underway. 1 . I N T R O D U C T I O N A (first-order) functional (applicative) program is a set of directed equations, used to compute by replacing instances of left-hand sides wi th the value of the r ighthand sides. For example, the following is a program for concatenating two lists of elements: where <= has the declarative meaning "is equal" (nil is the empty list and • denotes the cons function). Given the term it wi l l compute the result of appending the two ele­ ments a and 6 in the list a.b.nil [with parentheses, that should be a(bnil)] to the front of the list c.d.e.nil. A logic program [Kowalski-74] is a set of Horn clauses used as a pattern-directed program that searches for output terms that satisfy a given goal for given input terms. In this paradigm, the append program would be expressed in two statements: *This research was supported in part by the National Science Foundation under Grant MCS 83-07755 and in part by the Israeli Ministry of Defense while the author was at the Department of Mathematics and Computer Science, Bar-Han University, RamatGan, Israel. where : — has the declarative meaning "is implied by" . Given a goal append (a b nil ,c d e nil ,Z), this program computes the appended list Z=a.b.c.d.e.nil. Throughout this paper we follow the convention of using lower case for constants and upper case for free (universally quantified) variables. A rewrite system is a set of directed equations (or equivalences) used as a nondeterministic patterndirected program that returns as output a simplified term equal to a given input term. (See [Huet-Oppen-80] for a survey.) For example, the following is a three-rule rewrite system for append: Rules may be applied in any order to any matching subterm unt i l no further applications are possible. Thus, applying the rules to the term append {a nil,nil), one gets either the rewrite sequence append (a nil ,nil) a append (nil ,nil) a nil, or (using only the second rule) simply append (a nil ,nil) a nil. Such pattern-directed functional programming is avail­ able in SASL [Turner-79], HOPE [Burstall-MacQueenSannella-80], OBJ2 [Futatsugi,et al.-84], rewrite languages [Hoffmann-O'Donnell-82], and PLANNER-like languages. In this paper, we describe how the Knuth-Bendix completion procedure [Knuth-Bendix-70] may be used to generate programs f rom their (equational) specifications. The completion procedure was originally suggested as a means of generating rewrite systems that can be used to decide the val id i ty of identities in equational theories. Given a finite set of equations and a monotonic, wellfounded ordering on terms, the procedure generates