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

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 [...

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 ...

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...

single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single implicative axioms, that is, axioms of the form ff = fi ) fl = ffi. For Abelian groups, an axiom of this type with five variables was given by Sholander [6]. If we allow one of f

Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices and Lattice-like Structures 9 4 The Rule (gL) 23 4.1 Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 4.2 Sample Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 5 Quasigroups 51 6 Semigroups 57 6.1 A Conjecture of Padmanabhan : : : : : : : : : : : : : : : : : : : 57 7 Groups 69 7.1 Self-Dual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List

Algebras, pages 263-- 297. Pergamon Press, Oxford, U.K., 1970. [24] K. Kunen. Single axioms for groups. J. Automated Reasoning, 9(3):291--308, 1992. [25] H. Lakser, R. Padmanabhan, and C. R. Platt. Subdirect decomposition of P/lonka sums. Duke Math. J., 39(3):485--488, 1972. [26] A. I. Mal'cev. Uber die Einbettung von assoziativen Systemen Gruppen I. Mat. Sbornik, 6(48):331--336, 1939. [27] B. Mazur. Arithmetic on curves. Bull. AMS, 14:207--259, 1986. [28] J. McCharen, R. Overbeek, and L. Wos. Complexity and related enhancements for automated theorem-proving programs. Computers and Math. Applic., 2:1--16, 1976. [29] J. McCharen, R. Overbeek, and L. Wos. Problems and experiments for and with automated theorem-proving programs. IEEE Trans. on Computers, C-25(8):773--782, August 1976. [30] W. McCune. Automated discovery of new axiomatizations of the left group and right group calculi. J. Automated Reasoning, 9(1):1--24, 1992. [31] W. McCune. Single axioms for groups and Abelian g...

) Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. for abelian groups [3], and ((z \Delta (x \Delta y) \Gamma1 ) \Gamma1 \Delta (z \Delta y \Gamma1 )) \Delta (y \Gamma1 \Delta y) \Gamma1 = x (4) for ordinary groups [2]. One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting ff=fi to ff \Delta fi \Gamma1 . Doing so gives a single axiom, but then \Delta is not product, and \Gamma1