Cut-free Sequent Calculus for S5

We aim at an exposition of some nonstandard cut-free Gentzen formalization for S5, called DSC (double sequent calculus). DSC operates on two types of sequents instead of one, and shifting of wffs from one side of sequent to another is regulated by special rules and subject to some restrictions. Despite of this apparent inconvenience it seems to be simpler than other, known Gentzen-style systems for S5. The number of additional, formal machinery is kept in reasonable bounds and proofs are quite simple. A by-product of the completeness theorem for this calculus is an automated proof-search procedure. The article is an alternation of the introductory part of the bigger work [4], where detailed treatment of completeness and decidability is presented. One of the strange features of S5 is lack of simple and natural characterization in terms of sequent calculi. Despite its nice algebraic, modeltheoretic and syntactic properties, S5 when formulated via Gentzen approach meets some difficulties. The simplest (and closest to the original Gentzen formalism for classical logic) sequent calculus for modal logics, due to Ohnishi and Matsumoto [6], in the case of S5 is not cut-free. Since then, many authors devised various sequent systems, which allow for cutelimination, but sometimes it is reached through serious modifications of basic Gentzen apparatus. Usually, the language of the system is essentially enriched by new constants (cf. Belnap [1] or Wansing [8]) or indices (cf. Kanger [5] or Wansing [9]), sometimes rules for non-modal connectives are changed (e.g. Sato [7]). The present, cut-free formalisation DSC is believed to be quite simple and not very deviant from the standard Gentzen format. Subformulaproperty is saved, but in a slightly generalized form (see below). Some