Abstract
AbstractA complete classification of the complexity of the local and global satisfiability problems for graded modal language over traditional classes of frames has already been established. By “traditional” classes of frames, we mean those characterized by any positive combination of reflexivity, seriality, symmetry, transitivity, and the Euclidean property. In this paper, we fill the gaps remaining in an analogous classification of the graded modal language with graded converse modalities. In particular, we show its NExpTime-completeness over the class of Euclidean frames, demonstrating this way that over this class the considered language is harder than the language without graded modalities or without converse modalities. We also consider its variation disallowing graded converse modalities, but still admitting basic converse modalities. Our most important result for this variation is confirming an earlier conjecture that it is decidable over transitive frames. This contrasts with the undecidability of the language with graded converse modalities.
Highlights
For many years, modal logic has been an active topic in many academic disciplines, including philosophy, mathematics, linguistics, and computer science
We say that a modal logic F (L∗) has the finite local model property if any formula of L which is satisfied in some world of some structure from F is satisfied in some world of a finite structure from F
We argue that the local and global satisfiability problems coincide for modal logics over Euclidean frames
Summary
Modal logic has been an active topic in many academic disciplines, including philosophy, mathematics, linguistics, and computer science. Our aim is to classify the complexity of the local (“in a world”) and global (“in all worlds”) satisfiability problems for all the logics obtained by combining any of the above languages with any class of frames from the so-called modal cube, that is, a class of frames characterized by any positive combination of the axioms of reflexivity (T), seriality (D), symmetry (B), transitivity (4), and the Euclidean property (5). In the case of non-graded two-way modal language, over most relevant classes of frames, tight complexity bounds for local and global satisfiability are known. We prove that the logics K5(◇≥, ≥) and D5(◇≥, ≥) are locally and globally NExpTimecomplete This is a higher complexity than the ExpTime-complexity of the language without graded modalities (Demri and de Nivelle 2005) and NP-complexity of the language without converse (Kazakov and Pratt-Hartmann 2009) over the same classes of frames. This work is an extended version of our conference paper (Bednarczyk et al . 2019)
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