Managing automatically formed mathematical theories

Authors:
Simon Colton;Pedro Torres;Paul Cairns;Volker Sorge
Affiliations:
Department of Computing, Imperial College, London, UK;Department of Computing, Imperial College, London, UK;UCL Interaction Centre, University College, London, UK;School of Computing, University of Birmingham, UK
Venue:
MKM'06 Proceedings of the 5th international conference on Mathematical Knowledge Management
Year:
2006

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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.