Archive for Mathematical Logic

, Volume 42, Issue 3, pp 221–243 | Cite as

The modal logic of the countable random frame

  • Valentin Goranko
  • Bruce Kapron

Abstract.

 We study the modal logic M Lr of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M Lr and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Valentin Goranko
    • 1
  • Bruce Kapron
    • 2
  1. 1.Department of Mathematics, Rand Afrikaans University, PO Box 524, Auckland Park 2006, Johannesburg, South AfricaZA
  2. 2.Department of Computer Science, University of Victoria, British Columbia, CanadaCA

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