AOL: A logic of Acting, Sensing, Knowing, and Only Knowing

This work is motivated by the existence of two useful but quite different knowledge representation formalisms, the situation calculus due to McCarthy, and the logic OL of only knowing due to Levesque. In this paper, we propose the logic AOL, which combines both approaches in a clean and natural way. We present a semantics for AOL which generalizes the semantics of OL to account for actions, and a sound and complete set of axioms for AOL which generalizes the Lin and Reiter foundational axioms of the situation calculus to account for only knowing. The logic is compatible with earlier work on knowledge and action in that the solution to the frame problem for knowledge proposed by Scherl and Levesque becomes now a theorem of AOL. We also demonstrate that the logic avoids certain anomalies present in related work by Lakemeyer. Finally we provide a mapping from OL into AOL such that a sentence of OL is valid iff its mapping is a theorem of AOL, thus providing, for the first time, a complete axiomatic characterization of OL.