Weighted automata and logics meet computational complexity

Complexity classes such as $\#P$, $\oplus P$, GapP, OptP, NPMV, or the class of fuzzy languages realised by polynomial-time fuzzy nondeterministic Turing machines, can all be described in terms of a class $\mathrm{NP}\left[S\right]$ for a suitable semiring S, defined via weighted Turing machines over S similarly as NP is defined in the unweighted setting. Other complexity classes can be lifted to the quantitative world as well, the resulting classes relating 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 implicit in the existing literature and suggests a systematic approach to the study of weighted complexity classes. An extension of $\mathrm{SAT}$ to weighted propositional logic is proved to be complete for $\mathrm{NP}\left[S\right]$ when S is finitely generated. Moreover, a class $\mathrm{FP}\left[S\right]$ is introduced for each semiring S as a counterpart to P, and the relations between $\mathrm{FP}\left[S\right]$ and $\mathrm{NP}\left[S\right]$ are considered.