Automatic theorem generation in plane geometry

Bagai, Rajiv; Shanbhogue, V.; Żytkow, Jan M.; Chou, Shang-Ching

doi:10.1007/3-540-56804-2_39

Cited by 13 publications

(10 citation statements)

References 5 publications

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“…We look at a (relatively) little-studied algebraic structure which we refer to as star algebras. 2 These algebras have a single axiom, which resembles associativity: ∀ x, y, z ((x * y) * z = y * (z * x)). …”

Section: Enabling Mathematical Discovery -Illustrative Resultsmentioning

confidence: 99%

“…This fascination began with Lenat's inspirational -but ultimately flawed 1 -approach to mathematical concept formation via the AM and Eurisko programs [15], which formed concepts in set and number theory. Following these early attempts, methods for theory formation in particular domains were implemented, e.g., plane geometry [2], number systems (such as Conway numbers) [21] and non-associative algebras [12]. Particular attention has been paid to graph theory, with Epstein's GT program [8] providing a generic model for theory formation, and Fajtlowicz's Graffiti program [9] producing many conjectures, the proofs/disproofs of which have led to publication in the mathematical literature.…”

Section: Introductionmentioning

confidence: 99%

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Managing Automatically Formed Mathematical Theories

Colton

Torres

Cairns

et al. 2006

Lecture Notes in Computer Science

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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.

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Section: Enabling Mathematical Discovery -Illustrative Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Managing Automatically Formed Mathematical Theories

Colton

Torres

Cairns

et al. 2006

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The diagrams involved points and lines and relations between the points and lines, such as a point being on a line or two lines being parallel. For example, a parallelogram and its diagonals, as in Figure 5 (taken from Bagai et al, 1993), could be described by stating that there were four ingredient points, A, B, C and D, six lines (one between each pair of distinct points) and two relations, namely that lines AB and CD were parallel and that lines AC and BD were parallel.…”

Section: Bagai E¹ A¸1s Systemmentioning

confidence: 99%

“…The discovery program developed by Rajiv Bagai et al (1993), worked in plane geometry by constructing idealized diagrams and proving theorems stating that certain diagrams could not be drawn. Each concept consisted of a set of "rst-order statements representing a diagram in plane geometry.…”

Section: Bagai E¹ A¸1s Systemmentioning

confidence: 99%

On the notion of interestingness in automated mathematical discovery

Colton¹,

Bundy²,

Walsh³

2000

International Journal of Human-Computer Studies

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We survey "ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they 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 di!erent 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 di!erent 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 a!ects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.

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“…With the third method, he discovered a generalisation of Simson's theorem. (Bagai et al, 1993) provide a more general approach to automated exploration in plane geometry, although it appears that no new theorems resulted from this work.…”

Section: Discovering Theoremsmentioning

confidence: 99%

Computational Discovery in Pure Mathematics

Colton

Lecture Notes in Computer Science

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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 Simon's prediction that a computer would discover and prove an important mathematical theorem.

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Automatic theorem generation in plane geometry

Cited by 13 publications

References 5 publications

Managing Automatically Formed Mathematical Theories

Managing Automatically Formed Mathematical Theories

On the notion of interestingness in automated mathematical discovery

Computational Discovery in Pure Mathematics

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