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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 . . . . . . . . . . . . . . . . . . . . . ...

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.

Classifying finite algebraic structures has been a major motivation behind much research in pure mathematics. Automated techniques have aided this process, but this has largely been at a quantitative level, e.g., to prove that there are no quasigroups of a given type for a given size, or to count the number of groups of a particular order. Classification theorems of a more qualitative nature are often more interesting. For example, Kronecker's classification of finite Abelian groups [1] states that every Abelian group, G, of size n can be expressed as a direct product of cyclic groups, G = C s1 \Theta \Delta \Delta \Delta \Theta C sm, where n = s

: In this report we give an overview of the development of our new Waldmeister prover for equational theories. We elaborate a systematic stepwise design process, starting with the inference system for unfailing Knuth--Bendix completion and ending up with an implementation which avoids the main diseases today's provers suffer from: overindulgence in time and space. Our design process is based on a logical three--level system model consisting of basic operations for inference step execution, aggregated inference machine, and overall control strategy. Careful analysis of the inference system for unfailing completion has revealed the crucial points responsible for time and space consumption. For the low level of our model, we introduce specialized data structures and algorithms speeding up the running system and cutting it down in size --- both by one order of magnitude compared with standard techniques. Flexible control of the mid--level aggregation inside the resulting prover is made po...

With the aid of automated reasoning techniques, we show that all previously known short single axioms for odd exponent groups are special cases of one general schema. We also demonstrate how to convert the proofs generated by an automated reasoning system into proofs understandable by a human. x0. Introduction. There are two eras in the history of single axioms for groups and varieties of groups. The early results, by Neumann and others [7], often produced single axioms which were larger than the minimal possible size, but which were constructed via some scheme which made them easy to verify by hand. Although not optimal, these results had the virtue that a person could conceptually grasp their proofs. The second era began with the advent of McCune's automated reasoning system OTTER [4]; now one could produce shorter and simpler single axioms, which often required much more complex verifications. Short single axioms for groups and some varieties of groups were found by McCune and Wos [...

It is well known that the n-ary morphisms defined on projective algebraic curves satisfy some strong local-to-global equational rules of derivation not satisfied in general by universal algebras. For example, every rationally defined group law on a cubic curve must be commutative. Here we extract from the geometry of curves a firstorder property (gL) satisfied by all morphisms defined on these curves such that the equational consequences known for projective curves can be derived automatically from a set of six rules (stated within the first-order logic with equality). First, the rule (gL) is implemented in the theorem-proving program Otter. Then we use Otter to automatically prove some incidence theorems on projective curves without any further reference to the underlying geometry or topology of the curves. AMS Subject Classification (1991). Primary: 68T15, 08B05. Secondary:14H52, 20N05. 1 Introduction The term "equational logic" refers to the study of various metalogical notions rel...

this paper, we show that the equational theory of TBAs is one-based. Our methods for finding a single identity for the theory of TBAs are interesting from two distinct points of view. First, from the algebraic, since TBAs enjoy both permutable and distributive congruences, they admit a single ternary polynomial p(x; y; z), the socalled Pixley polynomial [1, p. 405]. We first find such a polynomial p(x; y; z) and use a technique of R. Padmanabhan and R. W. Quackenbush [7] to construct a single identity for the equational theory in question. This is done in Section 2. Second, from the viewpoint of automated reasoning, we use the program Otter to discover new single identities based upon the results of the algebraic view. Actually we obtain here three new identities--- shorter in length than those obtained by the formal algebraic process of Section 2---each characterizing the equational theory of TBAs. The relevant Otter proofs are also included. 2 The Algebraic View

. A method for building finite models is proposed. It combines enumeration of the set of interpretations on a finite domain with strategies in order to prune significantly the search space. The main new ideas underlying our method are to benefit from symmetries and from the information extracted from the structure of the problem and from failures of model verification tests. The algorithms formalizing the approach are given and the standard properties (termination, completeness, and soundness) are proven. The method can deal with first-order logic with equality. In contrast to existing ones, it does not require to transform the initial problem into a normal form and can be easily extended to other logics. Experimental results and comparisons with related works are reported. 1. Introduction The capital importance of the notion of "model" in Logic was naturally inherited by Automated Deduction, where, since the very beginning, the use of models has been recognized as an useful technique...

We study equations of the form (ff = x) which are single axioms for groups of exponent 4, where ff is a term in product only. Every such ff must have at least 9 variable occurrences, and there are exactly three such ff of this size, up to variable renaming and mirroring. These terms were found by an exhaustive search through all terms of this form. Automated techniques were used in two ways: to eliminate many ff by verifying that (ff = x) true in some non-group, and to verify that the group axioms do indeed follow from the successful (ff = x). We also present an improvement on Neumann's scheme for single axioms for varieties of groups. x0. Introduction. If n 1 is an integer, a group of exponent n is a group in which x n is the identity for all elements x. We study equations of the form (ff = x) which are single axioms for groups of exponent n, where ff is a term in product only. Note that in our definition of "exponent n", we do not require that n is the smallest exponent, so, for ...