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A guide to completeness and complexity for modal logics of knowledge and belief. (English) Zbl 0762.68029
Summary: We review and re-examine possible-worlds semantics for propositional logics of knowledge and belief with three particular points of emphasis:
(1) we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief,
(2) we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system, and
(3) we discuss how notions of common knowledge and distributed knowledge among a group of agents fit into the possible-worlds framework.
As far as complexity is concerned, we show, among other results, that while the problem of deciding satisfiability of an $$S5$$ formula with one agent is $$NP$$-complete, the problem for many agents in PSPACE-complete. Adding a distributed knowledge operator does not change the complexity, but once a common knowledge operator is added to the language, the problem becomes complete for exponential time.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 68T30 Knowledge representation 03B45 Modal logic (including the logic of norms)
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