2024/04/27 by Guillermo Badía, Badia, Guillermo, Manfred Droste +5 · 2 citations
Computer Science · #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge
paper · pdf · doi:10.48550/arxiv.2404.17784
openalex publication_date 2024/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fagin's seminal result characterizing NP in terms of existential second-order logic started the fruitful field of descriptive complexity theory. In recent years, there has been much interest in the investigation of quantitative (weighted) models of computations. In this paper, we start the study of descriptive complexity based on weighted Turing machines over arbitrary semirings. We provide machine-independent characterizations (over ordered structures) of the weighted complexity classes NP[S], FP[S], FPLOG[S], FPSPACE[S], and FPSPACEpoly[S] in terms of definability in suitable weighted logics for an arbitrary semiring S. In particular, we prove weighted versions of Fagin's theorem (even for arbitrary structures, not necessarily ordered, provided that the semiring is idempotent and commutative), the Immerman--Vardi's theorem (originally for P) and the Abiteboul--Vianu--Vardi's theorem (originally for PSPACE). We also address a recent open problem proposed by Eiter and Kiesel.