Weighted Automata and Logics Meet Computational Complexity
Peter Kostolányi
TL;DR
This work extends classical computational complexity into the quantitative realm by formulating an overarching framework based on semirings and weighted Turing machines. It defines $\mathbf{NP}[S]$ as the class of power series realizable in polynomial time over a semiring $S$, and introduces $\mathbf{FP}[S]$ as its deterministic counterpart; the paper then proves $\mathsf{SAT}[S]$ and $\mathsf{WTMSAT}[S]$ are $\mathbf{NP}[S]$-complete for finitely generated $S$, generalizing the Cook–Levin paradigm to weighted logics. It also analyzes the structural relationship between $\mathbf{FP}[S]$ and $\mathbf{NP}[S]$, including how these classes behave under semiring homomorphisms and factor semirings, and provides explicit separations (e.g., $\mathbf{FP}[S] \neq \mathbf{NP}[S]$ for some $S$ via PAL). Overall, the paper builds a cohesive program connecting weighted automata, weighted logics, and quantitative complexity, with implications for descriptive complexity and the study of complexity beyond languages.
Abstract
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.