On the notion of interestingness in automated mathematical discovery

Abstract We survey five 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 different 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 different 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 affects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.

[1]  F. K. Hanna,et al.  AM: A Case Study in AI Methodology , 1984, Artif. Intell..

[2]  Herbert A. Simon,et al.  Scientific discovery: compulalional explorations of the creative process , 1987 .

[3]  Elizabeth D. Mynatt.  The Writing on the Wall , 1999, INTERACT.

[4]  Graham J. Williams.  Evolutionary Hot Spots Data Mining - An Architecture for Exploring for Interesting Discoveries , 1999, PAKDD.

[5]  S. Blinkhorn.  The writing is on the wall , 1993, Nature.

[6]  Toby Walsh,et al.  Automatic Concept Formation in Pure Mathematics , 1999, IJCAI.

[7]  Neil R. Smalheiser,et al.  Artificial Intelligence An interactive system for finding complementary literatures : a stimulus to scientific discovery , 1995 .

[8]  Siemion Fajtlowicz,et al.  On conjectures of Graffiti , 1988, Discret. Math..

[9]  Douglas B. Lenat,et al.  EURISKO: A Program That Learns New Heuristics and Domain Concepts , 1983, Artif. Intell..

[10]  Pat Langley,et al.  The Computer-Aided Discovery of Scientific Knowledge , 1998, Discovery Science.

[11]  Douglas B. Lenat,et al.  Why AM and EURISKO Appear to Work , 1984, Artif. Intell..

[12]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[13]  Jan M. Zytkow,et al.  Automatic Theorem Generation in Plane Geometry , 1993, ISMIS.

[14]  Howard J. Hamilton,et al.  Heuristic for Ranking the Interestigness of Discovered Knowledge , 1999, PAKDD.

[15]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[16]  Susan L. Epstein.  On the Discovery of Mathematical Theorems , 1987, IJCAI.

[17]  Douglas B. Lenat,et al.  AM, an artificial intelligence approach to discovery in mathematics as heuristic search , 1976 .

[18]  Robert E. Kennedy,et al.  Tau numbers, natural density, and Hardy and Wright's theorem 437 , 1990 .

[19]  E. Vald.  Principles of human-computer collaboration for knowledge discovery in science , 1999 .

[20]  Kenneth W. Haase.  Discovery Systems , 1986, ECAI.

[21]  Raúl E. Valdés-Pérez,et al.  Machine Discovery in Chemistry: New Results , 1995, Artif. Intell..

[22]  Michael H. Sims,et al.  Discovering mathematical operator definitions , 1989, ICML 1989.

[23]  Susan L. Epstein.  Knowledge representation in mathematics: a case study in graph theory , 1983 .

[24]  Jean-Pierre Serre,et al.  Fermat ’ s Last Theorem , 2017 .

[25]  Susan L. Epstein.  Learning and discovery: one system's search for mathematical knowledge , 1988, Comput. Intell..

[26]  Martin Stacey,et al.  Scientific Discovery: Computational Explorations of the Creative Processes , 1988 .

[27]  Simon Colton.  HR - Automatic Concept Formation in Finite Algebras , 1998, AAAI/IAAI.

[28]  F. H. Bennett.  Genetic Programming : Biologically Inspired Computation that Exhibits Creativity in Solving Non-Trivial Problems , 1999 .

[29]  Wei-Min Shen,et al.  Functional transformations in AI discovery systems , 1988, [1988] Proceedings of the Twenty-First Annual Hawaii International Conference on System Sciences. Volume III: Decision Support and Knowledge Based Systems Track.

[30]  A. Kitchen,et al.  Knowledge based systems in artificial intelligence , 1985, Proceedings of the IEEE.

[31]  Simon Colton,et al.  Refactorable Numbers - A Machine Invention , 1999 .

[32]  David H. Bailey.  Finding new mathematical identities via numerical computations , 1998, SGNM.

[33]  W. McCune.  A Davis-Putnam program and its application to finite-order model search: Quasigroup existence problems , 1994 .

[34]  T. Nickles.  Scientific discovery, case studies , 1980 .

[35]  Jacques Wainer,et al.  Automatic theory formation in graph theory , 1999, J. Braz. Comput. Soc..

[36]  T. Kuhn,et al.  The Structure of Scientific Revolutions. , 1964 .

[37]  Gary L. Miller,et al.  On the nlog n isomorphism technique (A Preliminary Report) , 1978, STOC.