Logical Characterizations of Weighted Complexity Classes

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 FPSPACE_poly[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 discuss a recent open problem proposed by Eiter and Kiesel. Recently, the above mentioned weighted complexity classes have been investigated in connection to classical counting complexity classes. Furthermore, several classical counting complexity classes have been characterized in terms of particular weighted logics over the semiring ℕ of natural numbers. In this work, we cover several of these classes and obtain new results for others such as NPMV, ⊕𝖯, or the collection of real-valued languages realized by polynomial-time real-valued nondeterministic Turing machines. Furthermore, our results apply to classes based on many other important semirings, such as the max-plus and the min-plus semirings over the natural numbers which correspond to the classical classes MaxP[O(log n)] and MinP[O(log n)], respectively.