Complexity classes such as $\#\mathbf{P}$, $\oplus\mathbf{P}$, $\mathbf{GapP}$, $\mathbf{OptP}$, $\mathbf{NPMV}$, or the class of fuzzy languages realised by polynomial-time fuzzy nondeterministic Turing machines, can all be described in terms of a class $\mathbf{NP}[S]$ for a suitable semiring $S$, defined via weighted Turing machines over $S$ similarly as $\mathbf{NP}$ is defined via the classical nondeterministic Turing machines. Other complexity classes of decision problems can be lifted to the quantitative world using the same recipe as well, and the resulting classes relate to the original ones in the same way as weighted automata or logics relate to their unweighted counterparts. The article surveys these too-little-known connexions between weighted automata theory and computational complexity theory implicit in the existing literature, suggests a systematic approach to the study of weighted complexity classes, and presents several new observations strengthening the relation between both fields. In particular, it is proved that a natural extension of the Boolean satisfiability problem to weighted propositional logic is complete for the class $\mathbf{NP}[S]$ when $S$ is a finitely generated semiring. Moreover, a class of semiring-valued functions $\mathbf{FP}[S]$ is introduced for each semiring $S$ as a counterpart to the class $\mathbf{P}$, and the relations between $\mathbf{FP}[S]$ and $\mathbf{NP}[S]$ are considered.

翻译：诸如 $\#\mathbf{P}$、$\oplus\mathbf{P}$、$\mathbf{GapP}$、$\mathbf{OptP}$、$\mathbf{NPMV}$ 等复杂性类，以及由多项式时间模糊非确定性图灵机实现的模糊语言类，均可通过为适当半环 $S$ 定义的类 $\mathbf{NP}[S]$ 来描述。该类通过 $S$ 上的加权图灵机定义，其方式类似于 $\mathbf{NP}$ 通过经典非确定性图灵机的定义。其他决策问题的复杂性类也可通过相同方法提升至定量范畴，所得类别与原类别的关系，类似于加权自动机或逻辑学与其非加权对应物之间的关系。本文综述了现有文献中隐含但鲜为人知的加权自动机理论与计算复杂性理论之间的联系，提出了研究加权复杂性类的系统化方法，并给出了若干强化这两个领域关联的新观察结果。特别地，本文证明了当 $S$ 为有限生成半环时，加权命题逻辑中布尔可满足性问题的自然扩展对于类 $\mathbf{NP}[S]$ 具有完备性。此外，针对每个半环 $S$，引入了一类半环值函数 $\mathbf{FP}[S]$ 作为类 $\mathbf{P}$ 的对应物，并探讨了 $\mathbf{FP}[S]$ 与 $\mathbf{NP}[S]$ 之间的关系。