Application of Theorem Proving to Problem Solving

This paper shows how an extension of the resolution proof procedure can be used to construct problem solutions. The extended proof procedure can solve problems involving state transformations. The paper explores several alternate problem representations and provides a discussion of solutions to sample problems including the "Monkey and Bananas" puzzle and the 'Tower of Hanoi" puzzle. The paper exhibits solutions to these problems obtained by QA3, a computer program bused on these theorem-proving methods. In addition, the paper shows how QA3 can write simple computer programs and can solve practical problems for a simple robot. Automatic theorem proving by the resolution proof procedure § represents perhaps the most powerful known method for automatically determining the validity of a statement of first-order logic. In an earlier paper Green and Raphael" illustrated how an extended resolution procedure can be used as a question answerer—e.g., if the statement (3x)P(x) can be shown to follow from a set of axioms by the resolution proof procedure, then the extended proof procedure will find or construct an x that satisfies P(x). This earlier paper (1) showed how one can axiomatize simple question-answering subjects, (2) described a question-answering program called QA2 based on this procedure, and (3) presented examples of simple question-answering dialogues with QA2. In a more recent paper" the author (1) presents the answer construction method in detail and proves its correctness, (2) describes the latest version of the program, QA3, and (3) introduces state-transformation methods into the constructive proof formalism. In addition to the question-answering applications illustrated in these earlier papers, QA3 has been used as an SRI robot 4 problem solver and as an automatic program writer. The purpose of this paper is This preprint is a preliminary version and is subject to modification prior to publication. References are listed at the end of this paper. twofold: (1) to explore the question of predicate calculus representation for state-transformation problems in general, and (2) to elaborate upon robot and program-writing applications of this approach and the mechanisms underlying them. Exactly how one can use logic and theorem proving for problem solving requires careful thought on the part of the user. .Judging from my experience, and that of others using QA2 and QA3, one of the first difficulties encountered is the representation of problems, especially state-transformation problems, by statements in formal logic. Interest has been shown in seeing several detailed examples that illustrate alternate methods …