A very hard log space counting class

Abstract

Consideration is given to the logarithmic space counting classes Hash L, opt-L, and span-L, which are defined analogously to their polynomial-time counterparts. Complete functions are obtained for these three classes in terms of graphs and finite automata. It is shown that Hash L and opt-L are both contained in NC/sup 2/, but that, surprisingly, span-L seems to be much harder counting class than Hash L and opt-L. It is demonstrated that span-L-functions can be computed in polynomial time if and only if P=NP=PH=P( Hash P), i.e if the class P( Hash P) and all the classes of the polynomial-time hierarchy are contained in P. This result follows from the fact that span-L and Hash P are very similar: span-L contained in Hash P, and any function in Hash P can be represented as a subtraction of two functions in span-L. Nevertheless, Hash P contained in span-L would imply NL=P=NP. An investigation is also conducted of various restrictions of the classes opt-L and span-L, and it is shown, e.g that if opt-L coincides with one of its restricted versions, then L=NL follows. >