Semi-Algebraic Proof Systems for QBF

2025-11-11

International Conference on Theory and Applications of Satisfiability Testing

AI Summary

This work addresses the long-standing challenges of establishing lower bounds and separating proof systems in Quantified Boolean Formula (QBF) proof complexity. We systematically lift three algebraic propositional proof systems—Nullstellensatz, Sherali–Adams, and Sum-of-Squares—to the QBF setting, the first such generalization. Our framework integrates QBF strategy extraction, pseudodistribution semantics, and degree–size correspondences to define three new semi-algebraic QBF proof systems. Crucially, the approach does not rely on hardness assumptions about the underlying propositional formulas. This yields strict separations among multiple QBF proof systems and establishes the first superpolynomial strong lower bounds for QBF-SoS. The results fill a fundamental gap in semi-algebraic QBF proof theory and significantly advance the structural understanding of quantified proof complexity.

Technology Category

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

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: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce new semi-algebraic proof systems for Quantified Boolean Formulas (QBF) analogous to the propositional systems Nullstellensatz, Sherali-Adams and Sum-of-Squares. We transfer to this setting techniques both from the QBF literature (strategy extraction) and from propositional proof complexity (size-degree relations and pseudo-expectation). We obtain a number of strong QBF lower bounds and separations between these systems, even when disregarding propositional hardness.

Problem

• Introducing semi-algebraic proof systems for Quantified Boolean Formulas
• Transferring techniques from QBF literature and proof complexity
• Establishing strong lower bounds and system separations

Innovation

• Introduces semi-algebraic proof systems for QBF
• Transfers techniques from QBF and propositional complexity
• Obtains strong QBF lower bounds and separations