Mathematical applications of inductive logic programming

摘要

The application of Inductive Logic Programming to scientific datasets has been highly successful. Such applications have led to breakthroughs in the domain of interest and have driven the development of ILP systems. The application of AI techniques to mathematical discovery tasks, however, has largely involved computer algebra systems and theorem provers rather than machine learning systems. We discuss here the application of the HR and Progol machine learning programs to discovery tasks in mathematics. While Progol is an established ILP system, HR has historically not been described as an ILP system. However, many applications of HR have required the production of first order hypotheses given data expressed in a Prolog-style manner, and the core functionality of HR can be expressed in ILP terminology. In Colton (2003), we presented the first partial description of HR as an ILP system, and we build on this work to provide a full description here. HR performs a novel ILP routine called Automated Theory Formation, which combines inductive and deductive reasoning to form clausal theories consisting of classification rules and association rules. HR generates definitions using a set of production rules, interprets the definitions as classification rules, then uses the success sets of the definitions to induce hypotheses from which it extracts association rules. It uses third party theorem provers and model generators to check whether the association rules are entailed by a set of user supplied axioms.