Low Sets and Closure Properties of Counting Function Classes
Yaroslav Ivanashev
TL;DR
This work analyzes lowness for counting-function complexity classes, precisely characterizing low classes for TotP and providing Low_f characterizations for #P and SpanP via the subfamilies UPSV_t and NPSV_t. It connects these low-function classes to familiar counting classes (#P, GapP, TotP, SpanP) through a network of conditional inclusions and equivalences, revealing that many closure properties under left composition with FP_+ are equivalent to collapses of standard hierarchies (e.g., PP = UP, PP = NP). A key technical tool is the FP-assembly lemma showing FP ∘ #P can be simulated with a single #P oracle call, enabling one-query-sp-sp results and broader closure insights. Overall, the paper provides a cohesive framework linking lowness, counting classes, and FP-left-composition, with implications for potential collapses of complexity classes.
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 = $Σ_{2}^{P}$. 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.