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Short Single Axioms for Boolean Algebra
 J. Automated Reasoning
, 2002
"... We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke tha ..."
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Cited by 21 (11 self)
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We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke than the ones we present. Automated deduction techniques were used for several different aspects of the work. Keywords: Boolean algebra, Sheffer stroke, single axiom 1. Background and
Automating the search for elegant proofs
 J. Automated Reasoning
"... The research reported in this article was spawned by a colleague’s request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far ..."
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Cited by 8 (5 self)
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The research reported in this article was spawned by a colleague’s request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far shorter proof is presented in detail through a sequence of experiments. Although clearly not an algorithm, the methodology is sufficiently general to enable its use for seeking elegant proofs regardless of the domain of study. In addition to (usually) being more elegant, shorter proofs often provide the needed path to constructing a more efficient circuit, a more effective algorithm, and the like. The methodology relies heavily on the assistance of McCune’s automated reasoning program OTTER. Of the aspects of proof elegance, the main focus here is on proof length, with brief attention paid to the type of term present, the number of variables required, and the complexity of deduced steps. The methodology is iterative, relying heavily on the use of three strategies: the resonance strategy, the hot list strategy, and McCune’s ratio strategy. These strategies, as well as other features on which the methodology relies, do exhibit tendencies that can be exploited in the search for shorter proofs and for other objectives. To provide some insight regarding the value of the methodology, I discuss its successful application to
Automated discovery of single axioms for ortholattices
 Algebra Universalis
, 2005
"... Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complemen ..."
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Cited by 4 (1 self)
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Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complementation. Proofs are omitted but are available in an associated technical report and on the Web. We used computers extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. 1.
Short equational bases for ortholattices
 Preprint ANL/MCSP10870903, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. ..."
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Cited by 3 (3 self)
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Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. Proofs are omitted but are available in an associated technical report. Computers were used extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. The notion of computer proof is addressed. 1
Experiments concerning the Automated Search for Elegant Proofs
 Technical Memorandum ANL/MCSTM221, Mathematics and Computer Science Division, Argonne National Laboratory
, 1997
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Automated Equational Deduction with Otter
, 1995
"... 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 a ..."
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Cited by 1 (1 self)
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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 Latticelike 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 SelfDual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List
Automated Deduction in Equational Logic and Geometry
, 1995
"... Algebras, pages 263 297. Pergamon Press, Oxford, U.K., 1970. [24] K. Kunen. Single axioms for groups. J. Automated Reasoning, 9(3):291308, 1992. [25] H. Lakser, R. Padmanabhan, and C. R. Platt. Subdirect decomposition of P/lonka sums. Duke Math. J., 39(3):485488, 1972. [26] A. I. Mal'cev. ..."
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Algebras, pages 263 297. Pergamon Press, Oxford, U.K., 1970. [24] K. Kunen. Single axioms for groups. J. Automated Reasoning, 9(3):291308, 1992. [25] H. Lakser, R. Padmanabhan, and C. R. Platt. Subdirect decomposition of P/lonka sums. Duke Math. J., 39(3):485488, 1972. [26] A. I. Mal'cev. Uber die Einbettung von assoziativen Systemen Gruppen I. Mat. Sbornik, 6(48):331336, 1939. [27] B. Mazur. Arithmetic on curves. Bull. AMS, 14:207259, 1986. [28] J. McCharen, R. Overbeek, and L. Wos. Complexity and related enhancements for automated theoremproving programs. Computers and Math. Applic., 2:116, 1976. [29] J. McCharen, R. Overbeek, and L. Wos. Problems and experiments for and with automated theoremproving programs. IEEE Trans. on Computers, C25(8):773782, August 1976. [30] W. McCune. Automated discovery of new axiomatizations of the left group and right group calculi. J. Automated Reasoning, 9(1):124, 1992. [31] W. McCune. Single axioms for groups and Abelian g...
SINGLE AXIOMS: WITH AND WITHOUT COMPUTERS
"... This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software ..."
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This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software