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. Key Words: Theorem proving, resolution, problem solving, automatic programming, program writing, robots, state transformations, question answering. 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

. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computer-aided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

Interactive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality. 1

Irrelevant clauses in resolution problems increase the search space, making it hard to find proofs in a reasonable time. Simple relevance filtering methods, based on counting function symbols in clauses, improve the success rate for a variety of automatic theorem provers and with various initial settings. We have designed these techniques as part of a project to link automatic theorem provers to the interactive theorem prover Isabelle. They should be applicable to other situations where the resolution problems are produced mechanically and where completeness is less important than achieving a high success rate with limited processor time. 1

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This paper shows how a question-answering system can use first-order logic as its language and an automatic theorem prover, based upon the resolution inference principle, as its deductive mechanism. The resolution proof procedure is extended to a constructive proof procedure. An answer construction algorithm is given whereby the system is able not only to produce yes or no answers but also to find or construct an object satisfying a specified condition. A working computer program, QA3, based on these ideas, is described. The performance of the program, illustrated by extended examples, compares favorably with several other question-answering programs. Methods are presented for solving state transformation problems. In addition to question-answering, the program can do automatic programming

We analyze the search efficiency of a number of common refutational theorem proving strategies for first-order logic. Search efficiency is concerned with the total number of proofs and partial proofs generated, rather than with the sizes of the proofs. We show that most common strategies produce search spaces of exponential size even on simple sets of clauses, or else are not sensitive to the goal. However, clause linking, which uses a reduction to propositional calculus, has behavior that is more favorable in some respects, a property that it shares with methods that cache subgoals. A strategy which is of interest for term-rewriting based theorem proving is the A-ordering strategy, and we discuss it in some detail. We show some advantages of A-ordering over other strategies, which may help to explain its efficiency in practice. We also point out some of its combinatorial inefficiencies, especially in relation to goal-sensitivity and irrelevant clauses. In addition, SLD-reso...

An extension of semantic resolution is proposed. It is also an extension of the set of support as it can be considered as a particular case of semantic resolution. It is proved sound and refutationally complete. The extension is based on our former method for model building. The approach uses constrained clauses (or c-clauses), i.e. couples [[clause : constraint]]. Two important new features are introduced with respect to semantic resolution. Firstly, the method builds its own (finite or infinite) models to guide the search or to stop it if the initial set of clauses is satisfiable. Secondly, instead of evaluating a clause in an interpretation it imposes conditions (coded in its rules) to force a c-clause not to be evaluated to true in the interpretation it builds. The extension is limited in this paper to binary resolution but generalizing it to nary-resolution should be straightforward. The prover implementing our method is an extension of OTTER and compares advantageously with it ...

When the Model Elimination (ME) procedure was rst proposed, a notion of lemma was put forth as a promising augmentation to the basic complete proof procedure. Here the lemmas that are used are also discovered by the procedure in the same proof run. Several implementations of ME now exist but only a 1970's implementation explicitly examined this lemma mechanism, with indi erent results. We report on the successful use of lemmas using the METEOR implementation of ME. Not only does the lemma device permit METEOR to obtain proofs not otherwise obtainable by METEOR, or any other ME prover not using lemmas, but some well-known challenge problems are solved. We discuss several of these more di cult problems, including two challenge problems for uniform general-purpose provers, where METEOR was rst in obtaining the proof. The problems are not selected simply to show o the lemma device, but rather to understand it better. Thus, we choose problems with widely di erent characteristics, including one where very few lemmas are created automatically, the opposite of normal behavior. This selection points out the potential of, and the problems with, lemma use. The biggest problem normally is the selection of appropriate lemmas to retain from the large number generated. 1