On the Complexity of Explicit Modal Logics

  • Roman Kuznets
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1862)


Explicit modal logic was introduced by S. Artemov. Whereas the traditional modal logic uses atoms ☐F with a possible semantics “F is provable”, the explicit modal logic deals with atoms of form t:F, where t is a proof polynomial denoting a specific proof of a formula F. Artemov found the explicit modal logic \( \mathcal{L}\mathcal{P} \) in this new format and built an algorithm that recovers explicit proof polynomials corresponding to modalities in every derivation in K. Gödel’s modal provability calculus \( \mathcal{S}4 \). In this paper we study the complexity of \( \mathcal{L}\mathcal{P} \) as well as the complexity of explicit counterparts of the modal logics \( \mathcal{K}, \mathcal{D}, \mathcal{T}, \mathcal{K}\mathcal{A}, \mathcal{D}4 \) found by V. Brezhnev. The main result: the satisfiability problem for each of these explicit modal logics belongs to the class ∑2 p 2 of the polynomial hierarchy. Similar problem for the original modal logics is known to be PSPACE-complete. Therefore, explicit modal logics have much better upper complexity bounds than the original modal logics.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Roman Kuznets
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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