  Recent research has examined algorithms to minimize robots' resource
footprints. The class of combinatorial filters (discrete variants of
widely-used probabilistic estimators) has been studied and methods for reducing
their space requirements introduced. 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 multiple 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 also corrects flaws
present in all prior filter reduction methods. In employing SAT, the algorithm
provides a promising basis for future practical development.
