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18
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.
Automatic Concept Formation in Pure Mathematics
"... The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best ..."
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Cited by 38 (28 self)
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The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best first search. This approachhasledHRtothediscoveryofinterestingnewmathematics and enables it to build theories from just the axioms of finite algebras.
A Strategy for Constructing New Predicates in First Order Logic
 In Proceedings of the Third European Working Session on Learning
, 1988
"... There is increasing interest within the Machine Learning community in systems which automatically reformulate their problem representation by defining and constructing new predicates. A previous paper discussed such a system, called CIGOL, and gave a derivation for the mechanism of inverting individ ..."
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Cited by 16 (6 self)
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There is increasing interest within the Machine Learning community in systems which automatically reformulate their problem representation by defining and constructing new predicates. A previous paper discussed such a system, called CIGOL, and gave a derivation for the mechanism of inverting individual steps in first order resolution proofs. In this paper we describe an enhancement to CIGOL's learning strategy which strongly constrains the formation of new concepts and hypotheses. The new strategy is based on results from algorithmic information theory. Using these results it is possible to compute the probability that the simplifications produced by adopting new concepts or hypotheses are not based on chance regularities within the examples. This can be derived from the amount of information compression produced by replacing the examples with the hypothesised concepts. CIGOL's improved performance, based on an approximation of this strategy, is demonstrated by way of the automatic "di...
HR  A System for Machine Discovery in Finite Algebras
 ECAI 98 Workshop Programme
, 1998
"... We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theo ..."
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Cited by 8 (0 self)
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We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory.
An Applicationbased Comparison of Automated Theory Formation and Inductive Logic Programming
 Linkoping Electronic Articles in Computer and Information Science (special issue: Proceedings of Machine Intelligence
, 2000
"... Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been ..."
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Cited by 5 (5 self)
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Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been applied to some applications in machine learning. We discuss HR's application to inducing de nitions from examples, scienti c discovery, problem solving and puzzle generation. For each problem, we look at how theory formation was applied, and mention some initial results from using HR.
Managing automatically formed mathematical theories
 In Proceedings of the 5th International Conference on Mathematical Knowledge Management
, 2006
"... Abstract. The HR system forms scientific theories, and has found particularly successful application in domains of pure mathematics. Starting with only the axioms of an algebraic system, HR can generate dozens of example algebras, hundreds of concepts and thousands of conjectures, many of which have ..."
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Cited by 4 (3 self)
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Abstract. The HR system forms scientific theories, and has found particularly successful application in domains of pure mathematics. Starting with only the axioms of an algebraic system, HR can generate dozens of example algebras, hundreds of concepts and thousands of conjectures, many of which have first order proofs. Given the overwhelming amount of knowledge produced, we have provided HR with sophisticated tools for handling this data. We present here the first full description of these management tools. Moreover, we describe how careful analysis of the theories produced by HR – which is enabled by the management tools – has led us to make interesting discoveries in algebraic domains. We demonstrate this with some illustrative results from HR’s theories about an algebra of one axiom. The results fueled further developments, and led us to discover and prove a fundamental theorem about this domain. 1
Intelligent Machinery and Mathematical Discovery
 Science
, 1999
"... All published research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described for the first time. The fundamental principle underlying ..."
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Cited by 4 (0 self)
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All published research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described for the first time. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecturemaking may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.
A Survey of Research on Automated Mathematical ConjectureMaking
 FAJTLOWICZ (EDITORS), AMERICAN MATHEMATICAL SOCIETY
, 2005
"... The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlyi ..."
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Cited by 2 (0 self)
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The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecturemaking are surveyed—the success of a conjecturemaking program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions.
On Progress in the Automation of Mathematical ConjectureMaking
, 2001
"... Several attempts have been made to automate mathematical conjecturemaking. ..."
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Several attempts have been made to automate mathematical conjecturemaking.
An Updated Survey of Research in Automated Mathematical ConjectureMaking
"... This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlyin ..."
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This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecturemaking may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.