On the notion of interestingness in automated mathematical discovery

Colton, Simon; Bundy, Alan; Walsh, Toby

doi:10.1006/ijhc.2000.0394

Cited by 53 publications

(37 citation statements)

References 24 publications

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“…There is a large community of researchers working to design systems that can carry out mathematical reasoning effectively; and there is a smaller, but significant, community trying to automate mathematical discovery and concept formation (see e.g. Colton et al, 2000). If there is any domain of scientific inquiry for which one might expect the philosophy of mathematics to play a supporting role, this is it.…”

Section: Theories Of Mathematical Understandingmentioning

confidence: 99%

Computers in Mathematical Inquiry

Avigad¹

2008

The Philosophy of Mathematical Practice

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Section: Theories Of Mathematical Understandingmentioning

confidence: 99%

Computers in Mathematical Inquiry

Avigad¹

2008

The Philosophy of Mathematical Practice

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“…3-8) we summarise some salient theories, and in Part III (pp. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] we consider some connections between them. (Readers who are familiar with the work of Peirce, Toulmin, Pollock, Lakatos and Walton may safely proceed straight to Part III.)…”

Section: 2mentioning

confidence: 99%

Five theories of reasoning: Interconnections and applications to mathematics

Pease

Aberdein

2013

LLP

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Abstract. The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce's development of abductive reasoning , and explore some connections between, as well as within, the theories. For instance, we investigate Peirce's abduction to deal with surprising situations in mathematics, represent Pollock's examples in terms of Toulmin's layout, discuss connections between Toulmin's layout and Walton's argumentation schemes, and suggest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove's [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community.

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“…For more details of some the initial measures we implemented in HR, see chapters 9 and 10 of Colton (2002b). For a discussion of the general notion of interestingness in automated mathematics, see (Colton et al, 2000c).…”

Section: Best-first Search Strategiesmentioning

confidence: 99%

Mathematical applications of inductive logic programming

Colton

Muggleton

2006

Mach Learn

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The application of Inductive Logic Programming to scientific datasets has been highly successful. Such applications have led to breakthroughs in the domain of interest and have driven the development of ILP systems. The application of AI techniques to mathematical discovery tasks, however, has largely involved computer algebra systems and theorem provers rather than machine learning systems. We discuss here the application of the HR and Progol machine learning programs to discovery tasks in mathematics. While Progol is an established ILP system, HR has historically not been described as an ILP system. However, many applications of HR have required the production of first order hypotheses given data expressed in a Prolog-style manner, and the core functionality of HR can be expressed in ILP terminology. In Colton (2003), we presented the first partial description of HR as an ILP system, and we build on this work to provide a full description here. HR performs a novel ILP routine called Automated Theory Formation, which combines inductive and deductive reasoning to form clausal theories consisting of classification rules and association rules. HR generates definitions using a set of production rules, interprets the definitions as classification rules, then uses the success sets of the definitions to induce hypotheses from which it extracts association rules. It uses third party theorem provers and model generators to check whether the association rules are entailed by a set of user supplied axioms.HR has been applied successfully to a number of predictive, descriptive and subgroup discovery tasks in domains of pure mathematics. We survey various applications of HR which have led to it producing number theory results worthy of journal publication, graph theory results rivalling those of the highly successful Graffiti program and algebraic results leading to novel classification theorems. To further promote mathematics as a challenge domain for ILP systems, we present the first application of Progol to an algebraic domain-we use Progol to find algebraic properties of quasigroups, semigroups and magmas (groupoids) of varying sizes which differentiate pairs of non-isomorphic objects. This development is particularly interesting because algebraic domains have been an important proving ground Mach Learn (2006) 64:25-64 for both deduction systems and constraint solvers. We believe that AI programs written for discovery tasks will need to simultaneously employ a variety of reasoning techniques such as induction, abduction, deduction, calculation and invention. We argue that mathematics is not only a challenging domain for the application of ILP systems, but that mathematics could be a good domain in which to develop a new generation of systems which integrate various reasoning techniques.

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On the notion of interestingness in automated mathematical discovery

Cited by 53 publications

References 24 publications

Computers in Mathematical Inquiry

Computers in Mathematical Inquiry

Five theories of reasoning: Interconnections and applications to mathematics

Mathematical applications of inductive logic programming

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