Disjunctions of Two Dependence Atoms

2025-08-22

AI Summary

This paper investigates the computational complexity of model checking disjunctions of dependence atoms. For this fragment of dependence logic, we establish the first complete trichotomy: NL-complete, LOGSPACE-complete, or first-order definable. Our approach integrates model-checking algorithms, analysis of small-model properties, and intricate reductions to characterize a coherence condition that determines the precise complexity class. As a key technical contribution, we identify a novel LOGSPACE-complete subclass of 2CNF formulas expressible via dependence atoms. The results forge a tight connection between dependence logic and database functional dependencies, precisely delineating the expressibility boundaries within AC⁰, LOGSPACE, and NL. This work provides the first systematic classification framework for the computational complexity of dependence logic, advancing both theoretical understanding and practical applicability in database theory and finite-model theory.

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Dependence logic is a formalism that augments the syntax of first-order logic with dependence atoms asserting that the value of a variable is determined by the values of some other variables, i.e., dependence atoms express functional dependencies in relational databases. On finite structures, dependence logic captures NP, hence there are sentences of dependence logic whose model-checking problem is NP-complete. In fact, it is known that there are disjunctions of three dependence atoms whose model-checking problem is NP-complete. Motivated from considerations in database theory, we study the model-checking problem for disjunctions of two unary dependence atoms and establish a trichotomy theorem, namely, for every such formula, one of the following is true for the model-checking problem: (i) it is NL-complete; (ii) it is LOGSPACE-complete; (iii) it is first-order definable (hence, in AC[0]). Furthermore, we classify the complexity of the model-checking problem for disjunctions of two arbitrary dependence atoms, and also characterize when such a disjunction is coherent, i.e., when it satisfies a certain small-model property. Along the way, we identify a new class of 2CNF-formulas whose satisfiability problem is LOGSPACE-complete.

Problem

• Classifying complexity of disjunctions of two dependence atoms
• Establishing trichotomy for model-checking problem complexity
• Characterizing coherence conditions for two-atom disjunctions

Innovation

• Classifies complexity of two-dependence-atom disjunctions
• Establishes trichotomy theorem for model-checking problems
• Identifies new LOGSPACE-complete 2CNF satisfiability class