Theoretical study of symmetries in propositional calculus and applications

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 domain. In many cases, the set of clauses contains several symmetries i.e. the set of clauses remains invariant under a permutation of variable names. We will show how we can shorten the proofs of such problems. We present an algorithm which detects the symmetries and explain how the symmetries are introduced and used in the following methods: Slri, Davis and Putnam and Semantic Evaluation. With symmetries we have got good results on many known problems such pigeon hole, Schur's lemma, Ramsey, the eight queen etc. The most interesting one is that we have been able to prove for the first time the unsatisfiability of the Ramsey problem for 17 vertices and 3 colors.