Logarithmic space counting classes

Abstract

We consider the logarithmic space counting classes #L, opt-L, and span-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 contained in NCsuper2, but that, surprisingly, span-L seems to be a much harder counting class than #L and opt-L. We demonstrate that span-L-functions can be computed in polynomial time if and only if P = NP = PH = P(#P), i.e. iff the class P(#P) and all the classes of the polynomial time hierarchy are contained in P. This result follows from the fact that span-L and #P are very similar: span-L C #P, and any function in #P can be represented as a subtraction of two functions in span-L. Nevertheless, #P C span-L would imply NL = P = NP. We furthermore investigate various restrictions of the classes opt-L and span-L, and show, e.g., that if opt-L coincides with one of its restricted versions, then L = NL follows.