Logics of knowledge based on possible-world semantics are u seful in many areas of knowledge representation and reasoning, ranging from security t o distributed computing to game theory. In these models, an agent is said to know a fact ϕ if ϕ is true in all the worlds she considers possible. While reasoning about knowledge with t his semantics has proved useful, as is well known, it suffers from what is known in the literature as the logical omniscience problem: under possible-world semantics, agents know all t autologies and know the logical consequences of their knowledge. While logical omniscience is certainly not always an issue, in many applications it is. For example, in the context of distributed computing, we are interested in polynomial-time algorithms, although in some cases the knowledge needed to p erform optimally may require calculations that cannot be performed in polynomial time (u nless P=NP) [Moses and Tuttle 1988]; in the context of security, we may want to reason about computationally bounded adversaries who cannot factor a large composite number, and thus cannot be logically omniscient; in game theory, we may be interested in the impac t of computational resources on solution concepts (for example, what will agents do if com puting a Nash equilibrium is difficult). Not surprisingly, many approaches for dealing with the logi cal omniscience problem have been suggested (see [Fagin, Halpern, Moses, and Vardi 1 995, Chapter 9] and [Moreno 1998]).