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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.
Shortest axiomatizations of implicational S4 and S5
 the Notre Dame Journal of Formal Logic (NDJFL
"... 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 short ..."
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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.
The Application of Automated Reasoning to Formal Models of Combinatorial Optimization
 Applied Mathematics and Computation
"... Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was late ..."
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Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The longterm objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies. 1 Introduction Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynami...
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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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...
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.
Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction ⋆
"... www.math.du.edu/~mkinyon www.math.du.edu/~petr ..."
Universitaet Leipzig
"... Throughout the twentieth century, the worlds of logic and mathematics were well aware of Hilbert’s twentythree problems and the challenge they offered. Although not known until very recently, there existed yet one more challenge offered by Hilbert, his twentyfourth problem. This problem focuses on ..."
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Throughout the twentieth century, the worlds of logic and mathematics were well aware of Hilbert’s twentythree problems and the challenge they offered. Although not known until very recently, there existed yet one more challenge offered by Hilbert, his twentyfourth problem. This problem focuses on finding simpler proofs, on the criteria for measuring simplicity, and on the ‘‘development of a theory of the method of proof in mathematics in general’’. Of the three themes of Hilbert’s twentyfourth problem, the first two are central to this article. We visit various areas of logic, showing that some of the studies of the masters are indeed strongly connected to this newly discovered problem. We also demonstrate that the use of an automated reasoning program (specifically, W. McCune’s OTTER) enables one to address this challenging problem. We offer questions that remain unanswered.
The Arrival of Automated Reasoning
"... For some, the object of automated reasoning is the design and implementation of a program that offers sufficient power to enable one to contribute new and significant results to mathematics and to logic, as well as elsewhere. One measure of success rests with the number and quality of the results ob ..."
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For some, the object of automated reasoning is the design and implementation of a program that offers sufficient power to enable one to contribute new and significant results to mathematics and to logic, as well as elsewhere. One measure of success rests with the number and quality of the results obtained with the assistance of the program in focus. A less obvious measure (heavily in focus here) rests with the ability of a novice, in the domain under investigation, to make significant contributions to one or more fields of science by relying heavily on a given reasoning program. For example, if one who is totally unfamiliar with the area of study but skilled in automated reasoning can discover with an automated reasoning program impressive proofs, previously unknown axiom dependencies, and far more, then the field of automated reasoning has indeed arrived. This article details such—how one novice, with much experience with W. McCune’s program OTTER but no knowledge of the domains under investigation, obtained startling results in the study of areas of logic that include the BCSK logic and various extensions of that logic.
Communications of the ACM (to appear 1997) Also Preprint ANL/MCSP6550397 Programs That Offer Fast, Flawless, Logical Reasoning*
"... Sherlock Holmes and Mr. Spock of Star Trek could reason logically and flawlessly, always. Some people you know have that ability, sometimes. Unfortunately, without perfect reasoning, diverse problems arise: � Bugs in computer programs. If a sort program places Sun before Intel, more than disappointm ..."
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Sherlock Holmes and Mr. Spock of Star Trek could reason logically and flawlessly, always. Some people you know have that ability, sometimes. Unfortunately, without perfect reasoning, diverse problems arise: � Bugs in computer programs. If a sort program places Sun before Intel, more than disappointment is experienced. � Flaws in chip design. One Pentium chip became famous because of a flaw. � Errors in mathematical proofs. Papers having a title of the form ‘‘On an Error by MacLane’ ’ are well remembered, but not with pleasure (at least by MacLane). How can the likelihood of the various cited disasters be reduced? One answer is automated reasoning. (For further information on automated reasoning at Argonne National Laboratory, see