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OTTER 3.3 Reference Manual
"... by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any a ..."
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by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago. ii
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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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
A Shortest 2Basis for Boolean Algebra in Terms of the Sheffer Stroke
 J. Automated Reasoning
, 2003
"... In this article, we present a short 2basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2basis can be shorter. We also prove that the new 2basis is unique (for its length) up to applications of commutativity. Our proof of the 2basis was found by using the method of p ..."
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In this article, we present a short 2basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2basis can be shorter. We also prove that the new 2basis is unique (for its length) up to applications of commutativity. Our proof of the 2basis was found by using the method of proof sketches and relied on the use of an automated reasoning program.
Conquering the Meredith Single Axiom
 J. Automated Reasoning
, 2000
"... For more than three and onehalf 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 t ..."
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For more than three and onehalf 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 twovalued sentential (or propositional) calculus and whose conclusion is the Lukasiewicz threeaxiom 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...
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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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
Axiomatizing the Skew Boolean Propositional Calculus
, 2007
"... Abstract. The skew Boolean propositional calculus (SBP C) is a generalization of the classical propositional calculus that arises naturally in the study of certain wellknown deductive systems. In this article, we consider a candidate presentation of SBP C and prove it constitutes a Hilbertstyle ax ..."
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Abstract. The skew Boolean propositional calculus (SBP C) is a generalization of the classical propositional calculus that arises naturally in the study of certain wellknown deductive systems. In this article, we consider a candidate presentation of SBP C and prove it constitutes a Hilbertstyle 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.
Heaps and Data Structures: A Challenge for Automated Provers
"... Software verification is one of the most prominent application areas for automatic reasoning systems, but their potential improvement is limited by shortage of good benchmarks. Current benchmarks are usually large but shallow, require decision procedures, or have soundness problems. In contrast, we ..."
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Software verification is one of the most prominent application areas for automatic reasoning systems, but their potential improvement is limited by shortage of good benchmarks. Current benchmarks are usually large but shallow, require decision procedures, or have soundness problems. In contrast, we propose a family of benchmarks in firstorder logic with equality which is scalable, relatively simple to understand, yet closely resembles difficult verification conditions stemming from realworld C code. Based on this benchmark, we present a detailed comparison of different heap encodings using a number of SMT solvers and ATPs. Our results led to a performance gain of an order of magnitude for the C code verifier VCC.
Learningassisted theorem proving with millions of lemmas
 Journal of Symbolic Computation
, 2015
"... Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and reused in later proofs by formal mathema ..."
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Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and reused in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be reused in later proofs. We show that in combination with learningbased relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical li
Learning From Previous Proof Experience: A Survey
, 1999
"... We present an overview of various learning techniques used in automated theorem provers. We characterize the main problems arising in this context and classify the solutions to these problems from published approaches. We analyze the suitability of several combinations of solutions for different app ..."
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We present an overview of various learning techniques used in automated theorem provers. We characterize the main problems arising in this context and classify the solutions to these problems from published approaches. We analyze the suitability of several combinations of solutions for different approaches to theorem proving and place these combinations in a spectrum ranging from provers using very specialized learning approaches to optimally adapt to a small class of proof problems, to provers that learn more general kinds of knowledge, resulting in systems that are less efficient in special cases but show improved performance for a wide range of problems. Finally, we suggest combinations of solutions for various proof philosophies.