Closure Properties and Characterizations of TotP

Yaroslav Ivanashev

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arXiv · 2504.20262

Closure Properties and Characterizations of TotP

Yaroslav Ivanashev

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Abstract

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+.

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Yaroslav Ivanashev. 2025-04-28. Closure Properties and Characterizations of TotP. https://arxiv.org/abs/2504.20262

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