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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

In this article, we present a short 2-basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2-basis can be shorter. We also prove that the new 2-basis is unique (for its length) up to applications of commutativity. Our proof of the 2-basis was found by using the method of proof sketches and relied on the use of an automated reasoning program.

For more than three and one-half decades beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems that have served well range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for two-valued sentential (or propositional) calculus and whose conclusion is the Lukasiewicz three-axiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps...

he research reported in this article was spawned by a colleague's request to find an elegant proof (of a s m theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request wa et by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far shorter m proof is presented in detail through a sequence of experiments. Although clearly not an algorithm, the ethodology is sufficiently general to enable its use for seeking elegant proofs regardless of the domain o c of study. In addition to (usually) being more elegant, shorter proofs often provide the needed path t onstructing a more efficient circuit, a more effective algorithm, and the like. The methodology relies f e heavily on the assistance of McCune's automated reasoning program OTTER. Of the aspects of proo legance, the main focus here is on proof length, with brief attention paid to the type of term present, , r the number of variables required, and the complexity of deduced...

We present a method for making use of past proof experience called flexible re-enactment (FR). FR is actually a search-guiding heuristic that uses past proof experience to create a search bias. Given a proof P of a problem solved previously that is assumed to be similar to the current problem A, FR searches for P and in the "neighborhood" of P in order to find a proof of A. This heuristic use of past experience has certain advantages that make FR quite profitable and give it a wide range of applicability. Experimental studies substantiate and illustrate this claim. This work was supported by the Deutsche Forschungsgemeinschaft (DFG). 2 1 INTRODUCTION 1 Introduction Automated deduction is essentially a search problem that gives rise to potentially infinite search spaces because of general undecidability. Despite these unfavorable conditions state-of-the-art theorem provers have gained a remarkable level of performance mainly due to (problem-specific) search-guiding heuristics and...

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

Abstract. The skew Boolean propositional calculus (SBP C) is a generalization of the classical propositional calculus that arises naturally in the study of certain well-known deductive systems. In this article, we consider a candidate presentation of SBP C and prove it constitutes a Hilbert-style axiomatization. The problem reduces to establishing that the logic presented by the candidate axiomatization is algebraizable in the sense of Blok and Pigozzi. In turn, this is equivalent to verifying four particular formulas are derivable from the candidate presentation. Automated deduction methods played a central role in proving these four theorems. In particular, our approach relied heavily on the method of proof sketches. 1.

Abstract. Shortest possible axiomatizations for the implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.