A Very Hard log-Space Counting Class

Abstract We consider the logarithmic-space counting and optimization classes #L, span-L, and opt-L, which are defined analogously to their polynomial-time counterparts. We obtain complete functions for these three classes in terms of graphs and finite automata. We show that #L and opt-L are both included in NC 2 , but that, surprisingly, span-L seems to be a much harder class than #L and opt-L. We demonstrate that span-L functions can be computed in polynomial time if and only if P (#P) and all the classes of the polynomial-time hierarchy are included in P. This result follows from the fact that span-L and #P are very similar: span-L ⊆ #P, and any function in #P can be represented as the difference of a function in FL and a function in span-L. Nevertheless, the inclusion #P ⊆ span-L would imply NL = P = NP . We, furthermore, investigate restrictions of the classes opt-L and span-L.

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