Logical omniscience via proof complexity

• Authors: Sergei Artemov;Roman Kuznets
• Venue: CSL'06 Proceedings of the 20th international conference on Computer Science Logic
• Year: 2006

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 Ki(F) (‘agent iknows 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.