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The tree of knowledge in action: Towards a common perspective
 In G. Governatori, I. Hodkinson, & Y. Venema (Eds.), Proceedings of advances in modal logic
, 2006
"... abstract. We survey a number of decidablity and undecidablity results concerning epistemic temporal logic. The goal is to provide a general picture which will facilitate the ‘sharing of ideas ’ from a number of different areas concerned with modeling agents in interactive social situations. ..."
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abstract. We survey a number of decidablity and undecidablity results concerning epistemic temporal logic. The goal is to provide a general picture which will facilitate the ‘sharing of ideas ’ from a number of different areas concerned with modeling agents in interactive social situations.
Universität Bremen
"... Abstract—We show that the satisfiability problem for the twodimensional extension K ×K of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulás from 2001. Our lower bound technique allows us to derive further lower bounds for manydimensional modal logics for which only e ..."
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Abstract—We show that the satisfiability problem for the twodimensional extension K ×K of unimodal K is nonelementary, hereby confirming a conjecture of Marx and Mikulás from 2001. Our lower bound technique allows us to derive further lower bounds for manydimensional modal logics for which only elementary lower bounds were previously known. We also derive nonelementary lower bounds on the sizes of FefermanVaught decompositions w.r.t. product for any decomposable logic that is at least as expressive as unimodal K. Finally, we study the sizes of FefermanVaught decompositions and formulas in Gaifman normal form for fixedvariable fragments of firstorder logic. I.
The Complexity of Decomposing Modal and FirstOrder Theories
"... We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional 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 kn ..."
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We study the satisfiability problem of the logic K2 = K ×K, i.e., the twodimensional 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 NEXPhardness 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 kNEXPcomplete, 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 lowerbound technique allows us to derive also nonelementary lower bounds for the twodimensional 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 FefermanVaught 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 threevariable fragment FO3 of firstorder logic, we obtain the following immediate corollaries: (i) the size of FefermanVaught decompositions with respect to disjoint sum are inherently nonelementary and (ii) equivalent formulas in Gaifman normal form are inherently nonelementary.