A Spectrum of Applications of Automated Reasoning

The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it offers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune’s program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include finding a first proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert’s twenty-fourth problem—which is extremely amenable to attack with the appropriate automated reasoning program—a problem concerned with proof simplification. The methodologies include those for seeking shorter proofs and for finding proofs that avoid unwanted lemmas or classes of term, a specific option for seeking proofs with smaller equational or formula complexity, and a different option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbertstyle axiomatic, including details that permit one sometimes to gain new insights. We include questions still open and challenges that merit consideration. 1 Background and Perspective In this article, we make a strong case for the use in diverse applications of an automated reasoning program by mathematicians and logicians. The basis consists of brief discussions ∗This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.