The HR Program for Theorem Generation

Abstract

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 [11], the HR system [1] — named after mathematicians Hardy and Ramanujan — performs theory formation in mathematical domains. It works by (i) using the MACE model generator [9] to generate objects of interest from axiom sets (ii) performing the concept formation and conjecture making itself and (iii) using the Otter theorem prover [8] 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 [10]. HR is a Java program available for download here: http://www.dai.ed.ac.uk/~simonco/research/hr.

This work is supported by EPSRC Grant GR/M98012.