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On the Notion of Interestingness in Automated Mathematical Discovery
 International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."
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Cited by 64 (25 self)
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We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
Otter: The CADE13 Competition Incarnations
 JOURNAL OF AUTOMATED REASONING
, 1997
"... This article discusses the two incarnations of Otter entered in the CADE13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter. ..."
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Cited by 45 (3 self)
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This article discusses the two incarnations of Otter entered in the CADE13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.
Experiments in automated deduction with condensed detachment
 in Proceedings of the Eleventh International Conference on Automated Deduction (CADE11), Lecture Notes in Artificial Intelligence
, 1992
"... This paper contains the results of experiments with several search strategies on 112 problems involving condensed detachment. The problems are taken from nine di erent logic calculi: three versions of the twovalued sentential calculus, the manyvalued sentential calculus, the implicational calculus ..."
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Cited by 23 (8 self)
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This paper contains the results of experiments with several search strategies on 112 problems involving condensed detachment. The problems are taken from nine di erent logic calculi: three versions of the twovalued sentential calculus, the manyvalued sentential calculus, the implicational calculus, the equivalential calculus, the R calculus, the left group calculus, and the right group calculus. Each problem was given to the theorem prover Otter and was run with at least three strategies: (1) a basic strategy, (2) a strategy with a more re ned method for selecting clauses on which to focus, and (3) a strategy that uses the re ned selection mechanism and deletes deduced formulas according to some simple rules. Two new features of Otter are also presented: the re ned method for selecting the next formula on which to focus, and a method for controlling memory usage. 1
Application of Automated Deduction to the Search for Single Axioms for Exponent Groups
 in Logic Programming and Automated Reasoning
, 1995
"... 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 obt ..."
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Cited by 8 (5 self)
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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...
OTTER experiments in a system of combinatory logic
 Journal of Automated Reasoning
, 1995
"... Abstract. This paper describes some experiments involving the automated theoremproving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in T ..."
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Cited by 8 (1 self)
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Abstract. This paper describes some experiments involving the automated theoremproving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in TRC. 1. Introduction. OTTER [5] is a resolution/paramodulation theoremproving program for firstorder logic with equality. It has been used successfully in several areas of logic and algebra [8], [9], [6], [7], [4]. In this paper we describe our experiments with OTTER in the system TRC
Automatic construction and verification of isotopy invariants
 IN PROC. OF IJCAR 2006, LNAI
, 2006
"... We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two ma ..."
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Cited by 5 (3 self)
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We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two major technical problems, namely generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on substructures. In addition, given the complexity of the theorems which verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an intricate interplay of computer algebra, model generation, theorem proving and satisfiability solving methods. To demonstrate the power of the approach, we generate an isotopic classification theorem for loops of size 6, which extends the previously known result that there are 22. This result was previously beyond the capabilities of automated reasoning techniques.
Computational Discovery in Pure Mathematics
"... Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body ..."
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Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathematics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simonâ€™s prediction that a computer would discover and prove an important mathematical theorem. 1
Automated Theorem Discovery: Future Direction for Theorem Provers
"... One obvious and important aspect of automated theorem proving is that the users know in advance which theorem they wish to prove. A possible future direction for theorem provers is to enable users to discover theorems which they were not necessarily aware of. We survey previous attempts at this ..."
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One obvious and important aspect of automated theorem proving is that the users know in advance which theorem they wish to prove. A possible future direction for theorem provers is to enable users to discover theorems which they were not necessarily aware of. We survey previous attempts at this and give a new demonstration of theorem generation using our HR program [10] in the domain of `antiassociative' algebras. We also suggest three applications where this functionality may prove useful, and discuss how this would add value to theorem provers.
Finding Simple Proofs In Logic Calculi
, 1998
"... The design of searchguiding heuristics for theorem provers centers on minimizing the time required to find any proof. Mathematicians, however, are also interested in simple proofs. Relevant simplicity criteria like proof length, for instance, hardly play a role in the design of heuristics. In th ..."
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The design of searchguiding heuristics for theorem provers centers on minimizing the time required to find any proof. Mathematicians, however, are also interested in simple proofs. Relevant simplicity criteria like proof length, for instance, hardly play a role in the design of heuristics. In this report we present heuristics designed to find simple proofs and empirically evaluate their performance in the area of logic calculi. The experiments demonstrate that significantly simpler proofs are found without incurring increased search effort in many cases. As a matter of fact, the search for simpler proofs very often succeeds faster than a search guided by a "standard" heuristic based on counting symbols. 1 Introduction Problems related to an inference rule called condensed detachment (CD)also known as substitution and detachmenthave piqued the attention of mathematicians [15, 6] and researchers in the field of automated deduction [11, 17, 9, 13, 18, 2, 16, 4] alike. CD i...
Basic Research Problems: The Problem of Automated Theorem Finding*
"... Abstract. This article is the twentyfifth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to identify appropriate properties to permit an automated reasoning program to find new and interesting theorems, in contr ..."
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Abstract. This article is the twentyfifth of a series of articles discussing various open research problems in automated reasoning. The problem proposed for research asks one to identify appropriate properties to permit an automated reasoning program to find new and interesting theorems, in contrast to proving conjectured theorems. Such programs are now functioning in many domains as valuable reasoning assistants. A sharp increase in their value would occur if they could also be used as colleagues to (so to speak) produce research on their own. Key words. Automated reasoning, theorem finding, unsolved research problem. Question: What properties can be identified to permit an automated reasoning program to find new and interesting theorems, as opposed to proving conjectured theorems? ( This question is the thirtyfirst of 33 problems proposed for research in [3] and will be referred to as Research Problem 31 throughout this article. All references to sections, chapters, test problems, and such also refer to [3].) The field of automated reasoning is an outgrowth of the field of automated theorem proving. In fact, the dominant activity in automated reasoning is still that of proving some (conjectured) theorem.