Ordered rewriting and confluence
CADE-10 Proceedings of the tenth international conference on Automated deduction
Term rewriting and all that
CTRS '94 Proceedings of the 4th International Workshop on Conditional and Typed Rewriting Systems
Random Testing in Isabelle/HOL
SEFM '04 Proceedings of the Software Engineering and Formal Methods, Second International Conference
Rippling: meta-level guidance for mathematical reasoning
Rippling: meta-level guidance for mathematical reasoning
Decompositions of Natural Numbers: From a Case Study in Mathematical Theory Exploration
SYNASC '07 Proceedings of the Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
Automated theory formation in mathematics
IJCAI'77 Proceedings of the 5th international joint conference on Artificial intelligence - Volume 2
Ascertaining Mathematical Theorems
Electronic Notes in Theoretical Computer Science (ENTCS)
Isabelle/HOL: a proof assistant for higher-order logic
Isabelle/HOL: a proof assistant for higher-order logic
Scheme-based synthesis of inductive theories
MICAI'10 Proceedings of the 9th Mexican international conference on Advances in artificial intelligence: Part I
Conjecture Synthesis for Inductive Theories
Journal of Automated Reasoning
AProVE 1.2: automatic termination proofs in the dependency pair framework
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Slothrop: Knuth-Bendix completion with a modern termination checker
RTA'06 Proceedings of the 17th international conference on Term Rewriting and Applications
Automating inductive proofs using theory exploration
CADE'13 Proceedings of the 24th international conference on Automated Deduction
Mathematical practice, crowdsourcing, and social machines
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
European collaboration on automated reasoning
AI Communications - ECAI 2012 Turing and Anniversary Track
Hi-index | 12.05 |
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 formulae 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 a 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 exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. 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.