Belief State Estimation for Planning via Approximate Logical Filtering and Smoothing

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. Unfortunately such filtering, though exact, can be intractable. To this end, we propose logical smoothing, a form of backwards reasoning that works in concert with 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 prove properties of our algorithms, and experimentally demonstrate their behaviour. Smoothing together with 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.