  Answer set programming (ASP) is a popular declarative programming paradigm
with various applications. Programs can easily have many answer sets that
cannot be enumerated in practice, but counting still allows quantifying
solution spaces. If one counts under assumptions on literals, one obtains a
tool to comprehend parts of the solution space, so-called answer set
navigation. However, navigating through parts of the solution space requires
counting many times, which is expensive in theory. Knowledge compilation
compiles instances into representations on which counting works in polynomial
time. However, these techniques exist only for CNF formulas, and compiling ASP
programs into CNF formulas can introduce an exponential overhead. This paper
introduces a technique to iteratively count answer sets under assumptions on
knowledge compilations of CNFs that encode supported models. Our anytime
technique uses the inclusion-exclusion principle to improve bounds by over- and
undercounting systematically. In a preliminary empirical analysis, we
demonstrate promising results. After compiling the input (offline phase), our
approach quickly (re)counts.
