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.

翻译：通过修改NP计算的接受条件所定义的复杂性类已被广泛研究。例如，类UP包含可由至多有一条接受路径的非确定性多项式时间图灵机（NPTM）求解的判定问题——等价于至多有一个解的NP问题，其在密码学中扮演重要角色，因为P≠UP与单向函数的存在性等价。本文定义并考察了若干此类类的变体，其中接受条件涉及NPTM的计算路径总数，而非接受路径数量。这一方向反映了计数类#P与TotP之间的关系，前者是计算NPTM接受路径数量的函数类，后者是计算NPTM总路径数量的函数类。#P由Valiant（1979）引入，是NP问题计数版本的经典研究类。TotP包含所有其判定版本属于P的#P中自可归约计数问题，其中包括非负积和式、#PerfMatch和#Dnf-Sat等著名#P完全问题，因而在可近似计数问题研究中具有重要地位。我们证明本文引入的几乎所有类都与其"#接受路径"可定义的对应类重合。基于此，我们为parity-P、Modkp、SPP、WPP、C=P和PP类提出了一类新的完全问题，这些完全问题通过精简归约由TotP完全问题定义。