Notes on the Space Requirements for Checking Satisfiability in Modal Logics

Recently, there has been growing attention to the space requirements of tableau methods (see for example [7], [1], [12]). We have proposed in [10] a method of reducing modal consequence relations to the global and local consequence relation of (polymodal) K. The reductions used there did however not establish good time complexity bounds. In this note we shall use reduction functions to obtain rather sharp space bounds. These bounds can also be applied to show completeness of ordinary tableau systems, which in turn yield space bounds that are slightly different from the ones derived by applying the reduction functions alone. It has been shown by Hudelmaier ([7]) that satisfiability in S4 is O(n log n)– space computable, while satisfiability in K and KT are O(n log n)–space computable. AnO(n log n)–space bound for KD has been obtained by Basin, Matthews and Viganò ([1]). Viganò ([15]) has shown that satisfiability in K4, KD4 and S4 is in O(n log n)–space. Nguyen has reduced these bounds to O(n log n) for K4, K4D and S4 in [12]. The bound for K has been improved to O(n) by Hemaspaandra in [5]. We shall deal here with an abstract method for obtaining space bounds, using reduction functions. Reduction functions have been introduced in [9] and further developed in [10]. These functions were used to obtain a number of folklore results on standard modal systems in a uniform way. Examples were the finite model property and interpolation, but also complexity. However, the bounds for local satisfiability obtained there were not good enough. Here we shall improve these results rather drastically. In some cases we shall obtain linear bounds (namely for K.D, K.alt1), while in other cases we shall obtain a bound which is a product of the number of subformulae and the modal depth. In some cases this is somewhat less than the best known result (it can be quadratic in the length), in others it is better. Typically, with the help of tableau methods we can also establish the O(n log n) space bound for most systems. The research for this paper has been made possible by a Heisenberg– grant by the Deutsche Forschungsgemeinschaft. I wish to express my thanks Advances in Modal Logic, Volume 4, 1–22. c © 2002, by World Scientific Publishing Co. Pte. Ltd.