In this paper, we define and study variants of several complexity classes of decision problems that are defined via some criteria on the number of accepting paths of an NPTM. In these variants, we modify the acceptance criteria so that they concern the total number of computation paths 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, thus providing an alternative model of computation for them. Moreover, for each of these classes, we present a novel family of complete problems, which are defined via TotP-complete problems. This way, we show that all the aforementioned classes have complete problems that are defined via counting problems whose existence version is in P, in contrast to the standard way of obtaining completeness results via counting versions of NP-complete problems. To the best of our knowledge, prior to this work, such results were known only for parity-P and C=P.

翻译：本文定义并研究了通过非确定性图灵机（NPTM）接受路径数判定问题若干复杂度类的变体。在这些变体中，我们修改了接受准则，使其关注计算路径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Valiant（1979）提出的NP问题计数版本的经典研究类，后者则计算NPTM的总路径数。TotP包含#P中所有判定版本属于P的自归约计数问题，其中涵盖#P完全问题如非负永久函数、#PerfMatch和#DNF-Sat，因此在可近似计数问题研究中具有重要地位。本文证明，所引入的几乎所有类都与其'#接受路径'可定义的对应类相一致，从而为这些类提供了替代计算模型。此外，我们为每个类呈现了通过TotP完全问题定义的全新完全问题族。通过这种方式，我们证明了上述所有类均具有通过存在性版本属于P的计数问题定义的完全问题，这与通过NP完全问题计数版本获得完全性结果的标准方法形成对比。据我们所知，在本文之前，此类结果仅对parity-P和C=P成立。