An optimal method for reasoning about actions

In this paper we propose a novel method for reasoning about actions and knowledge that has optimal computational complexity. In our approach, we use two different extensions of public announcement logic (PAL). The first, PALA, extends PAL with assignments, and the second, that we call PALAT, extends PAL with assignments and tests; both are equally expressive. First, we encode Scherl&Levesque’s solution to the frame problem with knowledge into PALAT, and show that this is a polynomial transformation. Then, by extending Lutz’ method for PAL satisfiability checking to PALA, we establish an optimal decision procedure for PALA. Our method runs in nondeterministic polynomial time for one agent, in polynomial space for multiple agents and in deterministic exponential time when common knowledge is involved.