Tractability through symmetries in propositional calculus

Abstract

Many propositional calculus problems — for example the Ramsey or the pigeon-hole problems — can quite naturally be represented by a small set of first-order logical clauses which becomes a very large set of propositional clauses when we substitute the variables by the constants of the domainD. In many cases the set of clauses contains several symmetries, i.e. the set of clauses remains invariant under certain permutations of variable names. We show how we can shorten the proof of such problems. We first present an algorithm which detects the symmetries and then we explain how the symmetries are introduced and used in the following methods: SLRI, Davis and Putnam and semantic evaluation. Symmetries have been used to obtain results on many known problems, such as the pigeonhole, Schur's lemma, Ramsey's, the eight queen, etc. The most interesting one is that we have been able to prove for the first time the unsatisfiability of Ramsey's problem (17 vertices and three colors) which has been the subject of much research.