A Sound and Complete Tableau Calculus for Reasoning about only Knowing and Knowing at Most

Abstract

We define a tableau calculus for the logic of only knowing and knowing at most ONℒ, which is an extension of Levesque's logic of only knowing Oℒ. The method is based on the possible-world semantics of the logic ONℒ, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ONℒ and Oℒ. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ONℒ.