Complexity Results for Some Logics of Inconsistent Belief

Modal logics of knowledge have been used to represent and reason about such varied topics as distributed and multi-agent systems, security protocols, and interactive proof systems. We investigate a class of modal logics for representing agents or systems with imperfect or inconsistent knowledge. These logics replace the usual closure under full conjunction for the ¤ operator with progressively weaker versions, and comprise a hierarchy with the traditional modal logic K at the top, and an infinite number of logics ordered by inclusion under it, all strictly stronger than N , the weakest monotonic modal logic. Fagin, Halpern, Moses, and Vardi have proposed a related framework of epistemic agents with multiple “frames of mind” (localreasoning structures), and have used N to represent such structures. The results here show that there are stronger logics applicable to local-reasoning contexts, suggesting that stronger forms of inference can be used to represent imperfect knowledge-based agents and protocols. Further, it is shown that the satisfiability question for each of these logics is PSPACE-complete, strictly harder than N , for which it is NP-complete. This also answers a conjecture of Vardi: the border between NPand PSPACE-hardness in modal-logical satisfiability problems is associated with conjunctive closure, however weak, and is defined by the K hierarchy as a whole. M. Allen (May 5, 2004) 1

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