A New Method to Obtain Termination in Backward Proof Search For Modal Logic S4

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It is well known that problem of cycles starts up in tableau- or sequent-based decision procedures for S4 and a number of other modal logics. Traditional techniques used to ensure termination of algorithms for backward proof search in such modal logics are based on loop check. Since unrestricted loop check requires quite involved implementation techniques, effective loop check methods have been proposed. These methods are mainly based using the notion of history that involves a compact information about some previous parts of backward proof search. In the article a new method to obtain termination in backward proof search for modal logic S4 is proposed. This method is based on loop check-free sequent calculus and does not require any form history. Using this method translation of sequents into a certain normal form is not utilized. Instead of histories, we use marks and indices with the help of which applications of modal rules (namely, transitivity and reflexivity rules) are restricted. Instead of an unrestricted transitivity rule in the usual sequent calculus for S4 several transitivity rules (corresponding to specific positive occurrences of the necessity modality) are introduced. The peculiarities of the introduced transitivity rules along with proposed complete strategy of their application allow us to eliminate loop check and to restrict backtracking in derivations. By relying on the constructed loop check-free sequent calculus a Pspace procedure for determination of termination of backward proof search in modal logic S4 is presented.