The Complexity of Decomposing Modal and First-Order Theories

We study the satisfiability problem of the logic K2 &equal; K × K—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 in K2 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 also allows us to derive 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 two immediate corollaries: the size of Feferman-Vaught decompositions with respect to disjoint sum are inherently nonelementary, and equivalent formulas in Gaifman normal form are inherently nonelementary. Our second main result consists in providing effective elementary (more precisely, doubly exponential) upper bounds for the two-variable fragment FO2 of first-order logic both for Feferman-Vaught decompositions and for equivalent formulas in Gaifman normal form.

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