Tableau calculus for only knowing and knowing at most

We present a tableau method for Levesque's logic of only knowing OL. More precisely, we define a tableau calculus for the logic of only knowing and knowing at most (ONL), which is an extension of OL. The method is based on the possible-world semantics of the logic ONC, 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. Such a 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 in both ONL. and OL.