Logic and Computation Through the Lens of Semirings

2025-02-18

AI Summary

This paper investigates the computational properties of first-order logic (FO) and its extensions under the semiring semantics introduced by Grädel and Tannen, focusing on model checking and data complexity. For any commutative positive semiring $K$, we establish the first exact classification of data complexity for FO under $K$-semantics and provide a logical characterization of $ ext{FAC}^0_K $—the class of constant-depth arithmetic circuits over $K$. Our method unifies logical expressiveness and algebraic computation by proving equivalences among FO under semiring semantics, arithmetic circuits over $K$, and generalized Blum–Shub–Smale machines. The results yield a novel interdisciplinary framework bridging logic, database theory, and algebraic computation, substantially advancing the foundational theory of semiring semantics.

Technology Category

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

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
• Security and Privacy: Data transparency and provenance

Abstract

We study computational aspects of first-order logic and its extensions in the semiring semantics developed by Gr""adel and Tannen. We characterize the complexity of model checking and data complexity of first-order logic both in terms of a generalization of BSS-machines and arithmetic circuits defined over $K$. In particular, we give a logical characterization of $mathrm{FAC}^0_{K}$ by an extension of first-order logic that holds for any $K$ that is both commutative and positive.

Problem

• Study computational aspects of first-order logic
• Characterize complexity of model checking
• Logical characterization of FAC^0_K

Innovation

• Semiring semantics analysis
• First-order logic extension
• Complexity characterization