On the Relationship Between Bisimulation and Combinatorial Filter Reduction

Combinatorial filters are discrete structures for modeling and reasoning about robotic systems. Such filters are of interest not only because of the potential for reduction of the computational power needed to execute the filter, but also for the insight they can sometimes provide into the information requirements of certain robotic tasks. It is known that the filter minimization problem -that is, for a given filter, to find a combinatorial filter with the minimal number of states among all filters with equivalent behavior-is NP-hard. Intuition might suggest that the well-known notion of bisimulation might be of direct use for this minimization problem. Indeed, the bisimilarity relation -the union of all bisimulation relations over the state space of the original filter-is an equivalence relation, and one might attempt to reduce a filter by merging states that are equivalent under this relation. This paper studies this relationship between bisimulation and combinatorial filter reduction. Specifically, we show that every filter minimization problem can be solved by computing a quotient of the input filter with some relation, but that for some filters, the bisimilarity relation is not the correct relation for this purpose. We also characterize the result of the bisimulation quotient operation as the solution to a different, stricter filter minimization problem, and identify several classes of filters for which a variant of bisimulation, called compatibility, can be used to minimize filters in polynomial time.