The paper gives exact characterizations of low languages and low functions for the counting classes TotP, #P, GapP, and SpanP, and links their closure under composition to collapses such as PP=UP and PP=NP.
Closure Properties and Characterizations of TotP
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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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Low Sets and Closure Properties of Counting Function Classes
The paper gives exact characterizations of low languages and low functions for the counting classes TotP, #P, GapP, and SpanP, and links their closure under composition to collapses such as PP=UP and PP=NP.