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计算接受条件所定义的复杂度类已被广泛研究。例如，类UP包含由最多具有一条接受路径的非确定性多项式时间图灵机（NPTM）——等价于最多一个解的NP问题——可判定的决策问题，该类在密码学中扮演重要角色，因为P≠UP等价于单向函数的存在性。本文定义并考察了若干此类变体，其接受条件涉及NPTM计算路径的总数而非接受路径数。这一方向反映了计数类#P与TotP之间的关系，前者是计数NPTM接受路径数的函数类（由Valiant于1979年引入），后者则是计数NPTM总路径数的函数类。TotP包含所有在#P中且其判定版本属于P的自归约计数问题，其中包括非负永久核、#完美匹配和#析取范式可满足性等著名#P完全问题，因而在可近似计数问题研究中具有重要意义。本文证明，所引入的几乎所有新类均与其基于接受路径数定义的对应类一致。基于此，我们为奇偶性-P、模kP、SPP、WPP、C=P和PP等类提出了一类新的完全问题，这些问题通过节俭归约由TotP完全问题定义。