On the power of counting the total number of computation paths of NPTMs

Complexity classes defined by modifying the acceptance condition of NP computations have been extensively studied. For example, the class UP, which contains decision problems solvable by non-deterministic polynomial-time Turing machines (NPTMs) with at most one accepting path -- equivalently NP problems with at most one solution -- has played a significant role in cryptography, since P=/=UP is equivalent to the existence of one-way functions. In this paper, we define and examine variants of several such classes where the acceptance condition concerns the total number of computation paths of an NPTM, instead of the number of accepting ones. This direction reflects the relationship between the counting classes #P and TotP, which are the classes of functions that count the number of accepting paths and the total number of paths of NPTMs, respectively. The former is the well-studied class of counting versions of NP problems, introduced by Valiant (1979). The latter contains all self-reducible counting problems in #P whose decision version is in P, among them prominent #P-complete problems such as Non-negative Permanent, #PerfMatch, and #Dnf-Sat, thus playing a significant role in the study of approximable counting problems. We show that almost all classes introduced in this work coincide with their '# accepting paths'-definable counterparts. As a result, we present a novel family of complete problems for the classes parity-P, Modkp, SPP, WPP, C=P, and PP that are defined via TotP-complete problems under parsimonious reductions.