Cover Combinatorial Filters and Their Minimization Problem.

A recent research theme has been the development of automatic methods to minimize robots' resource footprints. In particular, the class of combinatorial filters (discrete variants of widely-used probabilistic estimators) has been studied and methods developed for automatically reducing their space requirements. This paper extends existing combinatorial filters by introducing a natural generalization that we dub cover combinatorial filters. In addressing the new---but still NP-complete---problem of minimization of cover filters, this paper shows that three of the concepts previously believed to be true about combinatorial filters (and actually conjectured, claimed, or assumed to be) are in fact false. For instance, minimization does not induce an equivalence relation. We give an exact algorithm for the cover filter minimization problem. Unlike prior work (based on graph coloring) we consider a type of clique-cover problem, involving a new conditional constraint, from which we can find more general relations. In addition to solving the more general problem, the algorithm we present also corrects flaws present in all prior filter reduction methods. The algorithm also forms a promising basis for practical future development as it involves a reduction to SAT.