Filtering denotes any method whereby an agent updates its belief state -- its knowledge of the state of the world -- from a sequence of actions and observations. In logical filtering, the belief state is a logical formula describing possible world states and the agent has a (possibly nondeterministic) logical model of its environment and sensors. This paper presents efficient logical filtering algorithms that maintain a compact belief state representation indefinitely, for a broad range of environment classes including nondeterministic, partially observable STRIPS environments and environments in which actions permute the state space. Efficient filtering is also possible when the belief state is represented using prime implicates, or when it is approximated by a logically weaker formula. The properties of domains that we identify can be used to launch further investigation into efficient projection, execution monitoring, planning, diagnosis, real-time control and reinforcement learning in partially observable domains.