Answer set programming (ASP) is a logic programming formalism used in various areas of artificial intelligence like combinatorial problem solving and knowledge representation and reasoning. It is known that enhancing ASP with function symbols makes basic reasoning problems highly undecidable. However, even in simple cases, state of the art reasoners, specifically those relying on a ground-and-solve approach, fail to produce a result. Therefore, we reconsider consistency as a basic reasoning problem for ASP. We show reductions that give an intuition for the high level of undecidability. These insights allow for a more fine-grained analysis where we characterize ASP programs as “frugal” and “non-proliferous”. For such programs, we are not only able to semi-decide consistency but we also propose a grounding procedure that yields finite groundings on more ASP programs with the concept of “forbidden” facts.