The HR Program for Theorem Generation
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Automated theory formation involves the production of objects of interest, concepts about those objects, conjectures relating the concepts and proofs of the conjectures. In group theory, for example, the objects of interest are the groups themselves, the concepts include element types, subgroup types, etc., the conjectures include implication and if-and-only-if conjectures and these become theorems if they are proved, non-theorems if disproved. Similar to Zhang’s MCS program , the HR system  — named after mathematicians Hardy and Ramanujan — performs theory formation in mathematical domains. It works by (i) using the MACE model generator  to generate objects of interest from axiom sets (ii) performing the concept formation and conjecture making itself and (iii) using the Otter theorem prover  to prove conjectures. In domains where Otter and MACE are effective, HR can produce large numbers of theorems for testing automated theorem provers (ATPs), or smaller numbers of prime implicates, which represent some of the fundamental facts in a domain. We explain how HR operates in §2 and give details of a representative session in §3. As discussed in §4, the applications of HR to automated reasoning include the generation of constraints for constraint satisfaction problems, the generation of lemmas for automated theorem proving, and the production of benchmark theorems for the TPTP library of test problems for ATP systems . HR is a Java program available for download here: http://www.dai.ed.ac.uk/~simonco/research/hr.
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- 1.S. Colton. Automated Theory Formation in Pure Mathematics. PhD thesis, Department of Artificial Intelligence, University of Edinburgh, 2000.Google Scholar
- 2.S. Colton, A. Bundy, and T. Walsh. Automatic identification of mathematical concepts. In Machine Learning: Proceedings of the 17th International Conference, 2000.Google Scholar
- 4.S. Colton and I Miguel. Constraint generation via automated theory formation. In Proceedings of CP-01, 2001.Google Scholar
- 5.S. Colton and G. Sutcliffe. Automatic Generation of Benchmark Problems for ATP Systems. In Proceedings of the 7th Symposium on AI and Mathematics, 2002.Google Scholar
- 6.Andreas Franke and Michael Kohlhase. Mathweb, an agent-based communication layer for distributed automated theorem proving. In CADE 16, 1999.Google Scholar
- 7.P. Jackson. Computing prime implicates incrementally. In CADE 11, 1992.Google Scholar
- 8.W. McCune. The OTTER user’s guide. Technical Report ANL/90/9, Argonne National Laboratories, 1990.Google Scholar
- 9.W. McCune. A Davis-Putnam program and its application to finite first-order model search. Technical Report ANL/MCS-TM-194, Argonne National Laboratories, 1994.Google Scholar
- 11.J. Zhang. MCS: Model-based conjecture searching. In CADE-16, 1999.Google Scholar