Scheme-based synthesis of inductive theories

Authors:
Omar Montano-Rivas;Roy McCasland;Lucas Dixon;Alan Bundy
Affiliations:
School of Informatics, University of Edinburgh;School of Informatics, University of Edinburgh;School of Informatics, University of Edinburgh;School of Informatics, University of Edinburgh
Venue:
MICAI'10 Proceedings of the 9th Mexican international conference on Advances in artificial intelligence: Part I
Year:
2010

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Abstract

We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on 'schemes', which are terms in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of the theory can be used not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory formation systems. We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory of natural numbers and a theory of lists.