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