Logical Filtering And Smoothing: State Estimation In Partially Observable Domains

State estimation is the task of estimating the state of a partially observable dynamical system given a sequence of executed actions and observations. In logical settings, state estimation can be realized via logical filtering, which is exact but can be intractable. We propose logical smoothing, a form of backwards reasoning that works in concert with approximated logical filtering to refine past beliefs in light of new observations. We characterize the notion of logical smoothing together with an algorithm for backwards-forwards state estimation. We also present an approximation of our smoothing algorithm that is space efficient. We prove properties of our algorithms, and experimentally demonstrate their behaviour, contrasting them with state estimation methods for planning. Smoothing and backwards-forwards reasoning are important techniques for reasoning about partially observable dynamical systems, introducing the logical analogue of effective techniques from control theory and dynamic programming.