OTTER and the Moufang identity problem

Abstract

This article provides additional evidence of the value of using an automated reasoning program as a research assistant. Featured is the use of Bill McCune's program OTTER to find proofs of theorems taken from the study of Moufang loops, but not just any proofs. Specifically, the proofs satisfy the property of purity. In particular, when given, say, four equivalent identities (which is the case in this article), one is asked to prove the second identity from the first, the third from the second, the fourth from the third, and the first from the fourth. If the proof that 1 implies 2 does not rely on 3 or 4, then by definition the proof is pure with respect to 3 and 4, or simply the proof is pure. If for the four identities one finds four pure proofs showing that 1 implies 2, 2 implies 3, 3 implies 4, and 4 implies 1, then by definition one has found a circle of pure proofs. By finding the needed twelve pure proofs, this article shows that there does exist a circle of pure proofs for the four equivalent identities for Moufang loops and for all orderings of the identities; however, for much of this article, the emphasis is on the first three identities. In addition — in part to promote the use of automated reasoning programs and to answer questions concerning the choice of options — featured here is the methodology that was employed and a discussion of some of the obstacles, some of which are subtle. The approach relies on paramodulation (which generalizes equality substitution), on demodulation, and — so crucial for attacking deep questions and hard problems — on various strategies, most important of which are the hot list strategy, the set of support strategy, and McCune's ratio strategy. To permit verification of the results presented here, extension of them, and application of the methodology to other unrelated fields, a sample input file and four proofs (relevant to a circle of pure proofs for the four identities) are included. Research topics and challenges are offered at the close of this article.