Satisfiability Problems on Sums of Kripke Frames

We consider the operation of sum on Kripke frames, where a family frames-summands is indexed by elements another frame. In many cases, modal logic sums inherits finite model property and decidability from summands [Babenyshev Rybakov 2010 ; Shapirovsky 2018 ]. this paper we show that, under general condition, satisfiability problem polynomial space Turing reducible to summands. particular, for logics in PSpace an immediate corollary semantic characterization logic.