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：复杂性类如$\#\mathbf{P}$、$\oplus\mathbf{P}$、$\mathbf{GapP}$、$\mathbf{OptP}$、$\mathbf{NPMV}$，或由多项式时间模糊非确定性图灵机实现的模糊语言类，均可通过适当的半环$S$上的类$\mathbf{NP}[S]$来描述，其定义方式与$\mathbf{NP}$通过经典非确定性图灵机定义类似，即基于$S$上的加权图灵机。其他判定问题的复杂性类也可通过相同方法提升至量化世界，所得类与原始类的关系，正如加权自动机或逻辑与其无加权对应物之间的关系。本文综述现有文献中隐晦提及的加权自动机理论与计算复杂性理论之间鲜为人知的关联，提出一种系统性研究加权复杂性类的方法，并呈现若干加强两领域联系的新发现。特别地，本文证明当$S$是有限生成半环时，布尔可满足性问题在加权命题逻辑中的自然扩展对于类$\mathbf{NP}[S]$是完全的。此外，针对每个半环$S$，引入半环值函数类$\mathbf{FP}[S]$作为类$\mathbf{P}$的对应，并探讨$\mathbf{FP}[S]$与$\mathbf{NP}[S]$之间的关系。