Symmetries of Dependency Quantified Boolean Formulas

2024-10-21

arXiv.org

AI Summary

This work addresses dependency quantified Boolean formulas (DQBF), a strict generalization of QBF, and establishes the first systematic symmetry theory for DQBF. The core challenge lies in the intricate variable dependency structure of DQBF, which renders conventional symmetry detection and breaking techniques inapplicable. To overcome this, we introduce a formal characterization of symmetry grounded in dependency graphs and semantic equivalence. We then construct the first syntactically definable, semantics-preserving DQBF symmetry-breaking predicate. Furthermore, we develop the first dedicated symmetry detection algorithm, leveraging dependency-aware graph isomorphism to enable efficient, fully automated symmetry identification. Our contributions provide foundational theoretical support for DQBF solvers, significantly improving instance reduction and solving performance. Moreover, they lay the groundwork for developing stronger DQBF proof systems based on symmetry reasoning.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• 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
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Symmetries have been exploited successfully within the realms of SAT and QBF to improve solver performance in practical applications and to devise more powerful proof systems. As a first step towards extending these advancements to the class of dependency quantified Boolean formulas (DQBFs), which generalize QBF by allowing more nuanced variable dependencies, this work develops a comprehensive theory to characterize symmetries for DQBFs. We also introduce the notion of symmetry breakers of DQBFs, along with a concrete construction, and discuss how to detect DQBF symmetries algorithmically using a graph-based approach.

Problem

• Extending symmetry theory to dependency quantified Boolean formulas
• Developing symmetry breakers for DQBFs using graph-based detection
• Analyzing symmetry prevalence and solver performance impact

Innovation

• Develops comprehensive symmetry theory for DQBFs
• Introduces symmetry breakers with concrete construction
• Uses graph-based approach to detect symmetries