@MISC{_thecomplexity, author = {}, title = {The Complexity of Decomposing Modal and First-Order Theories}, year = {} }

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Abstract

We study the satisfiability problem of the logic K2 = K ×K, i.e., the two-dimensional variant of unimodal logic, where models are restricted to asynchronous products of two Kripke frames. Gabbay and Shehtman proved in 1998 that this problem is decidable in a tower of exponentials. So far the best known lower bound is NEXP-hardness shown by Marx and Mikulás in 2001. Our first main result closes this complexity gap: We show that satisfiability inK2 is nonelementary. More precisely, we prove that it is k-NEXP-complete, where k is the switching depth (the minimal modal rank among the two dimensions) of the input formula, hereby solving a conjecture of Marx and Mikulás. Using our lower-bound technique allows us to derive also nonelementary lower bounds for the two-dimensional modal logics K4 ×K and S52 ×K for which only elementary lower bounds were previously known. Moreover, we apply our technique to prove nonelementary lower bounds for the sizes of Feferman-Vaught decompositions with respect to product for any decomposable logic that is at least as expressive as unimodal K, generalizing a recent result by the first author and Lin. For the three-variable fragment FO3 of first-order logic, we obtain the following immediate corollaries: (i) the size of Feferman-Vaught decompositions with respect to disjoint sum are inherently nonelementary and (ii) equivalent formulas in Gaifman normal form are inherently nonelementary.