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Logical Omniscience Via Proof Complexity

  • Sergei Artemov
  • Roman Kuznets
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)

Abstract

The Hintikka-style modal logic approach to knowledge contains a well-known defect of logical omniscience, i.e., the unrealistic feature that an agent knows all logical consequences of her assumptions. In this paper, we suggest the following Logical Omniscience Test (LOT): an epistemic system E is not logically omniscient if for any valid in E knowledge assertion \(\mathcal{A}\) of type ‘Fis known,’ there is a proof of F in E, the complexity of which is bounded by some polynomial in the length of \(\mathcal{A}\). We show that the usual epistemic modal logics are logically omniscient (modulo some common complexity assumptions). We also apply LOT to evidence-based knowledge systems, which, along with the usual knowledge operator K i (F) (‘agent i knows F’), contain evidence assertions t:F (‘t is a justification for  F’). In evidence-based systems, the evidence part is an appropriate extension of the Logic of Proofs LP, which guarantees that the collection of evidence terms t is rich enough to match modal logic. We show that evidence-based knowledge systems are logically omniscient w.r.t. the usual knowledge and are not logically omniscient w.r.t. evidence-based knowledge.

Keywords

Modal Logic Proof System Epistemic Logic Derivation Tree Axiom Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sergei Artemov
    • 1
  • Roman Kuznets
    • 1
  1. 1.CUNY Graduate CenterNew York CityUSA

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