A Contraction-Free Sequent Calculus for S4

Abstract

Theorem proving in the modal logic S4 is notoriously difficult, because in conventional sequent style calculi for this logic lengths of deductions are not bounded in terms of the length of their endsequent. This means that the usual depth first search strategy for backwards construction of deductions of given sequents may give rise to infinite search paths and is not guaranteed to terminate. Thus using such a search strategy prevents us not only from obtaining a decision procedure for the logic in question, but even from arriving at a complete proof procedure. There are two well known approaches for overcoming this problem: both approaches rely on the fact that all formulas which occur in a deduction are subformulas of the end-sequent and that out of these formulas one may only form finitely many “essentially different” sequents. Thus although there is no bound on the length of all deductions of a given sequent, we know that for any given sequent there is a number such that if the sequent is deducible at all, then it has a deduction of length smaller than this number. Hence by only considering deductions of appropriately bounded length one may obtain a decision procedure. But due to the fact that using such a procedure one is forced to construct many redundant inferences—one will, for instance, have to consecutively apply the same inference many times—this approach is considered rather inefficient.