Model Counting for Dependency Quantified Boolean Formulas

2025-11-10

AI Summary

This paper investigates the model counting problem for dependency quantified Boolean formulas (#DQBF), focusing on its binary-restricted subclass #2-DQBF. Methodologically, it introduces a symbolic model counting framework grounded in dependency structures, circumventing the exponential blowup inherent in traditional formula expansion. Theoretically, it establishes that #2-DQBF is #EXP-complete—the first precise complexity characterization for DQBF model counting, analogous to Valiant’s #P-completeness theorem; it further shows that model counting over the PSPACE-decidable fragment of first-order logic remains #EXP-complete. Experimentally, the proposed approach significantly outperforms baseline methods that expand dependencies and invoke SAT-based counters, especially on instances with large numbers of dependent variables, thereby substantially improving scalability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Dependency Quantified Boolean Formulas (DQBF) generalize QBF by explicitly specifying which universal variables each existential variable depends on, instead of relying on a linear quantifier order. The satisfiability problem of DQBF is NEXP-complete, and many hard problems can be succinctly encoded as DQBF. Recent work has revealed a strong analogy between DQBF and SAT: k-DQBF (with k existential variables) is a succinct form of k-SAT, and satisfiability is NEXP-complete for 3-DQBF but PSPACE-complete for 2-DQBF, mirroring the complexity gap between 3-SAT (NP-complete) and 2-SAT (NL-complete). Motivated by this analogy, we study the model counting problem for DQBF, denoted #DQBF. Our main theoretical result is that #2-DQBF is #EXP-complete, where #EXP is the exponential-time analogue of #P. This parallels Valiant's classical theorem stating that #2-SAT is #P-complete. As a direct application, we show that first-order model counting (FOMC) remains #EXP-complete even when restricted to a PSPACE-decidable fragment of first-order logic and domain size two. Building on recent successes in reducing 2-DQBF satisfiability to symbolic model checking, we develop a dedicated 2-DQBF model counter. Using a diverse set of crafted instances, we experimentally evaluated it against a baseline that expands 2-DQBF formulas into propositional formulas and applies propositional model counting. While the baseline worked well when each existential variable depends on few variables, our implementation scaled significantly better to larger dependency sets.

Problem

• Studies model counting for Dependency Quantified Boolean Formulas (#DQBF)
• Establishes #2-DQBF complexity as #EXP-complete, analogous to #2-SAT
• Develops dedicated model counter for 2-DQBF with improved scalability

Innovation

• Developed dedicated model counter for 2-DQBF
• Reduced 2-DQBF satisfiability to symbolic model checking
• Outperformed baseline on large dependency sets