This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...

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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.

Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easy-to-find proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ

Problems stemming from the study of logic calculi in connection with an inference rule called "condensed detachment" are widely acknowledged as prominent test sets for automated deduction systems and their search guiding heuristics. It is in the light of these problems that we demonstrate the power of heuristics that make use of past proof experience with numerous experiments. We present two such heuristics. The first heuristic attempts to re-enact a proof of a proof problem found in the past in a flexible way in order to find a proof of a similar problem. The second heuristic employs "features" in connection with past proof experience to prune the search space. Both these heuristics not only allow for substantial speed-ups, but also make it possible to prove problems that were out of reach when using so-called basic heuristics. Moreover, a combination of these two heuristics can further increase performance. We compare our results with the results the creators of Otter obtained with t...

Many non-classical logics can be axiomatized by means of Hilbert Systems. Reasoning in Hilbert Systems, however, is extremely inefficient. Most inference methods therefore use the semantics of a logic in one kind or another to get more efficiency. In this paper a combination of Hilbert style and semantic reasoning is proposed. It is particularly tailored for cases where either the semantics of some operators is not known, or it is second-order, or it is just too complicated to handle, or flexibility in experimenting with different versions of a logic is required. First-order predicate logic is used as a meta-logic for combining the Hilbert part with the semantics part. Reasoning is done in a (theory) resolution framework. The basic method is applicable to many different (monotonic propositional) non-classical logics. It can, however, be improved by treating particular formulae in a special way, as rewrite rules, as theory unification or t...

We present new results in axiomatic group theory obtained by using automated deduction programs. The results include single axioms, some with the identity and others without, for groups of exponents 3, 4, 5, and 7, and a general form for single axioms for groups of odd exponent. The results were obtained by using the programs in three separate ways: as a symbolic calculator, to search for proofs, and to search for counterexamples. We also touch on relations between logic programmingand automated reasoning. 1 Introduction A group of exponent n is a group in which for all elements x, x n is the identity e. Groups of exponent 2, xx = e, are also called Boolean groups. A single axiom for an equational theory is an equality from which the entire theory can be derived by equational reasoning. We are concerned with single axioms for groups of exponent n, n 2. B. H. Neumann [6, p.83] gives a general form for single axioms for certain subvarieties of groups, including exponent groups. The a...

. At CADE-10 Ross Overbeek proposed a two-part contest to stimulate and reward work in automated theorem proving. The first part consists of seven theorems to be proved with resolution, and the second part of equational theorems. Our theorem provers Otter and its parallel child Roo proved all of the resolution theorems and half of the equational theorems. This paper represents a family of entries in the contest. Key Words. Automated theorem proving, resolution, paramodulation, Knuth-Bendix completion, strategy. 1 Introduction The Conference on Automated Deduction (CADE) has been for nearly twenty years a meeting where both theoreticians and practitioners present their work. Feeling perhaps that the conference was becoming dominated by the theoreticians, Ross Overbeek proposed at CADE-10 in 1990 a contest to stimulate work on the implementation and use of automated theorem-proving systems. The challenge was to prove a set of theorems, and do so with a uniform approach. That is, one w...

We examine different possibilities of coupling saturation-based theorem provers by exchanging positive/negative information. We discuss which positive or negative information is well-suited for cooperative theorem proving and show in an abstract way how this information can be used. Based on this study, we introduce a basic model for cooperative theorem proving. We present theoretical results regarding the exchange of positive/negative information as well as practical methods and heuristics that allow for a gain of efficiency in comparison with sequential provers. Finally, we report on experimental studies conducted in the areas condensed detachment, unfailing completion, and superposition. The author was supported by the Deutsche Forschungsgemeinschaft (DFG). 2 1 INTRODUCTION 1 Introduction In general, automated theorem proving is based on the solution of search problems which usually comprise huge search spaces. Thus, it is on the one hand necessary to develop fast algorithms a...

We present a cooperation concept for automated theorem provers that is based on a periodical interchange of selected results between several incarnations of a prover. These incarnations differ from each other in the search heuristic they employ for guiding the search of the prover. Depending on the strengths' and weaknesses of these heuristics different knowledge and different communication structures are used for selecting the results to interchange. Our concept is easy to implement and can easily be integrated into already existing theorem provers. Moreover, the resulting cooperation allows the distributed system to find proofs much faster than single heuristics working alone. We substantiate these claims by two case studies: experiments with the DiCoDe system that is based on the condensed detachment rule and experiments with the SPASS system, a prover for first order logic with equality based on the superposition calculus. Both case studies show the improvements by our cooperatio...