Decidability and Complexity via Mosaics of the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders

Abstract

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Satisfiability problems on sums of Kripke frames

“…• Sums and products with linear indices In the linear case, modal products are typically (highly) undecidable [RZ01]. However, the modal satisfiability problem on the lexicographic squares of dense unbounded linear orders is in NP [BM13]. This positive result seems to be scalable according to the following observation: in many cases, ϕ is satisfiable in sums over linear (quasi)orders iff ϕ is satisfiable in such sums that the length of indices is bounded by #ϕ.…”

“…This section elaborates earlier works [Sha08,Sha05] (in particular, our new results significantly generalize and simplify Theorems 22 and 35 in [Sha08]). In Section 5 we discuss modifications of the sum operation: (iterated) lexicographic sums of frames, which are important in the context of provability logics [Bek10]; lexicographic products of frames, earlier studied in [Bal09,Bal10,BM13,BFD16]; the operation of refinement of modal logics introduced in [BR10]. Further results and directions are discussed in Section 6.…”

About the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders: A New Study of Its Computability

“…It has been used for defining the semantical basis of different languages designed for time representation and temporal reasoning from the perspective of non-standard analysis [1,2]. In [3,4], the temporal logic of the lexicographic products of unbounded dense linear orders has been considered, its complete axiomatization has been given and its computability has been studied. The purpose of this paper is give a new proof of the membership in N P of the satisfiability problem of the lexicographic products of linear temporal logics.…”