arXiv subjects

Yaroslav Ivanashev

Publications and source records attributed to Yaroslav Ivanashev.

3 recordsLinked to original sources

arXiv2026

Low Sets and Closure Properties of Counting Function Classes

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 = $Σ_{2}^{\text{P}}$. 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.

cs.CC↗

arXiv2026

On the Complexity of Computing Outputs of a Metric Turing Machine

The classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ contain functions that compute the median solution for certain types of problems. In this paper, for these classes we introduce analogous classes of functions that compute the k-th solution, where k is an order function that depends on the input. We prove that the classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ are polynomial-time 1-Turing inter-reducible with the corresponding classes, where the order function is from FP or FP$^{\text{#P}}$. For MedP we also prove that it coincides with the corresponding classes, where the order function is from FP or #P. For several inclusions between function classes we give equivalent inclusions between language classes. In particular, we establish inclusion relations between MaxP and median classes MidP, MedP, and $\small{\overline{\text{MedP}}}$. We also prove that NPSV$_{\text{t}} \subseteq$ MaxP $\subseteq$ FP$^{\text{NP}}$ and both inclusions are proper if and only if NP $\neq$ coNP.

cs.CC↗

arXiv2025

Closure Properties and Characterizations of TotP

The class TotP consists of functions that count the number of all paths of a nondeterministic polynomial-time Turing machine. In this paper, we give a predicate based definition of TotP, analogous to a standard definition of #P. From a new characterization of TotP it follows that many well known #P problems belong to TotP, and TotP = #P if and only if P = NP. We show that TotP has several closure properties of #P and GapP, and also properties that are not known to hold for #P and GapP. We also prove that the closure of TotP under left composition with FP+ is equivalent to TotP = FP+ and P = PP, and give examples of FP+-functions such that if TotP is closed under composition with them, then it is closed under composition with FP+.

cs.CC↗