Low Sets and Closure Properties of Counting Function Classes

TL;DR

This paper investigates the properties of low and closure classes within counting function classes, establishing new equalities, characterizations, and implications for complexity class inclusions and collapses.

Contribution

It proves that Low(TotP) equals P, characterizes low function classes for several counting classes, and links class closures to collapses in the polynomial hierarchy.

Findings

Low(TotP) = P

Low_f(#P) = UPSV_t and Low_f(SpanP) = NPSV_t

Closure under left composition with FP_+ implies class collapse

Abstract

A language L is low for a relativizable complexity class C, if C$^L$ = 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$_f$(#P) = UPSV$_t$ and Low$_f$(SpanP) = NPSV$_t$. We establish the inclusion relations between NPSV$_t$, UPSV$_t$, 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 = $\Sigma_{2}^{P}$. For the class #P we prove that its closure under left composition with FP$_+$ is equivalent to #P = UPSV$_t$, and for SpanP this closure is equivalent to SpanP = NPSV$_t$. 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.