Low Sets and Closure Properties of Counting Function Classes

Abstract

A language L is low for a relativizable complexity class C, if C$^{\text{L}}$ = C. For the classes #P, GapP, and SpanP the exact low classes of languages are known: Low(#P) = UP $\cap$ coUP, Low(GapP) = SPP, and Low(SpanP) = NP $\cap$ coNP. In this paper, we prove that Low(TotP) = P, and give characterizations of low function classes for #P, GapP, TotP, and SpanP. In particular, we prove that Low$_{\text{f}}$(#P) = UPSV$_{\text{t}}$ and Low$_{\text{f}}$(SpanP) = NPSV$_{\text{t}}$. We establish the inclusion relations between NPSV$_{\text{t}}$, UPSV$_{\text{t}}$, and the counting function classes by giving for each of these inclusions an equivalent inclusion between language classes. We also prove that SpanP $\subseteq$ GapP if and only if NP $\subseteq$ SPP, and the inclusion GapP$_+$ $\subseteq$ SpanP implies PH = $\Sigma_{2}^{\text{P}}$. For the class #P we prove that its closure under left composition with FP$_+$ is equivalent to #P = UPSV$_{\text{t}}$, and for SpanP this closure is equivalent to SpanP = NPSV$_{\text{t}}$. For the classes #P, GapP, TotP, and SpanP we summarize the known results and show that each of these classes is closed under left composition with FP$_{+}$ if and only if it collapses to its low class of functions. We also prove that a NPTM with a #P oracle can always make at most one query to the oracle without changing the number of accepting paths.