Loop Detection in Prolog in the Case of Possible Backtracking

A method will be proposed for the detection of some periodic loops during the execution of Prolog programs, and an implementation of this method will be described. The method is an extension of another one presented in the paper [3]. Namely, that method was intended for the detection of loops only during depth-first search, whereas the new one will work essentially in the same way in that case, but the combination of depth-first search with backtracking will be no more an obstacle. As in [3], we shall consider only Prolog programs consisting of Horn clauses and goals consisting of atomic formulas. For describing our loop detection method and specifying the class of the detected loops, we use a mathematical model of Prolog execution with backtracking. The model takes into account only such features of the execution process which are of interest for our purpose; due to this, it is much simpler than, for example, the D. H. D. Warren Prolog Engine (cf. [1], pp. 447-471). From now on, we suppose a Prolog program P is given. Let C1, . . . , Cm be the clauses of this program. We shall consider objects called marked goals. By definition, a marked goal is an ordered pair , where k is some of the numbers 0, 1, . . . , m, and G is an ordinary goal (if G is the goal consisting of the atomic formulas A1, . . . , Al, then will be denoted also by k − A1, . . . , Al. ). Roughly speaking, the number k is intended to forbid using clauses Ci with i ≤ k for a resolution with G. The number of the atomic formulas in G will be called the length of . The beginnings of are, by definition, the marked goals , where G′ is a beginning of G. Two marked goals and will be called to be essentially the same if k = k′, and G coincides with G′ up to some renaming of variables. Any finite sequence of marked goals will be called an execution state (such a sequence (M1, M2, . . . , Md) will be denoted also by M1 M2 . . . Md , and thus the one-term sequence (M1) will be identified with its only member M1). Intuitively speaking, an execution state is intended to represent some current content of a stack needed for the execution of P . We assume that the direction from top to bottom in the stack is represented by the one from left