Fagin's Theorem for Semiring Turing Machines

2025-07-24

AI Summary

This paper addresses quantitative computational complexity modeling over commutative semirings by introducing and systematically studying a novel complexity class, $mathrm{NP}_{mathrm{inf}}(mathcal{R})$. Methodologically, it establishes the first Fagin-style logical characterization theorem for semiring Turing machines, using weighted existential second-order logic ($existsmathrm{SO}^{mathcal{R}}$) to precisely capture their computational power and characterize the class of semiring formal power series languages. The main contributions are: (1) a complete logical characterization of $mathrm{NP}_{mathrm{inf}}(mathcal{R})$; (2) a precise delineation of its hierarchical relationships with classical NP and counting complexity classes—including $#mathrm{P}$ and $mathrm{PP}$; and (3) a unifying reconstruction and substantial generalization of Eiter & Kiesel (2023), thereby providing the first bidirectional logic-computation bridge for quantitative complexity theory grounded in semiring semantics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In recent years, quantitative complexity over semirings has been intensively investigated. An important problem in this context is to connect computational complexity with logical expressiveness. In this paper we improve on the model of emph{Semiring Turing Machines} (distinct from so called weighted Turing machines) introduced by Eiter & Kiesel (Semiring Reasoning Frameworks in AI and Their Computational Complexity, emph{J. Artif. Intell. Res.}, 2023). Our central result is a Fagin-style theorem for a new quantitative complexity class using a suitable weighted logical formalism. We show that the quantitative complexity class that we call NPnewinf{$mathcal{R}$}, where $mathcal{R}$ is a commutative semiring, can be captured using a version of weighted existential second-order logic that allows for predicates interpreted as semiring-annotated relations. This result provides a precise logical characterization of the power series that form the class NPnewinf{$mathcal{R}$}. We also give the exact relation between Eiter & Kiesel's version of NP, called NPoldinf{$mathcal{R}$}, and the class NPnewinf{$mathcal{R}$}. Incidentally, we are able to recapture all the complexity results by Eiter & Kiesel (2023) in our new model, connecting a quantitative version of NP to various counting complexity classes.

Problem

• Connects computational complexity with logical expressiveness over semirings
• Introduces a new quantitative complexity class NPnewinf{R}
• Provides logical characterization of power series in NPnewinf{R}

Innovation

• Introduces Semiring Turing Machines model
• Proves Fagin-style theorem for new complexity class
• Uses weighted existential second-order logic