Automated reasoning and machine learning

Abstract

This dissertation introduces new theorem-proving strategies and uses these strategies to solve a wide variety of difficult problems requiring logical reasoning. It also shows how to use theorem-proving to solve the problem of learning mathematical concepts. Our first algorithm constructs formulas called Craig interpolants from the refutation proofs generated by contemporary theorem-provers using binary resolution, paramodulation, and factoring. This algorithm can construct the formulas needed to learn concepts expressible in the full first-order logic from examples of the concept. It can also find sentences which distinguish pairs of nonisomorphic finite structures. We then apply case analysis to solve hard problems such as the zebra problem, the pigeonhole problem, and the stable marriage problem. The case analysis technique we use is the first to be fully compatible with resolution and rewriting and powerful enough to solve these problems. Our primary new theorem-proving strategies generate subgoals and efficient sets of rules. We show how to divide problems into smaller parts with intermediate goals by reversing logical implications. We solve these subdivided parts by discovering efficient subsets of rules or by generating efficient new rules. We apply these and other new search strategies to solve difficult problems such as the 15-puzzle, central solitaire, TopSpin, Rubik's cube, and masterball. Our strategies apply universally to all such problems and can solve them quite efficiently: the 15-puzzle, Rubik's cube and masterball can all be done in 300 seconds. Finally we apply our search strategies to solve real-world problems such as sorting, solving equations and inverting nonsingular matrices.