Polynomial Prenexing of QBFs with Non-Monotone Boolean Operators

2025-06-14

AI Summary

Existing prenexing methods for Quantified Boolean Formulas (QBFs) containing non-monotonic Boolean operators (e.g., ↔, ⊕) fail to preserve quantifier depth in polynomial time, causing theoretical and practical bottlenecks. Method: We propose the first polynomial-time prenex transformation that strictly preserves quantifier depth. Our approach combines structural induction and algebraic operator decomposition with quantifier scope contraction and equivalence replacement lemmas to convert any QBF with non-monotonic operators into an equivalent standard prenex form using only ¬, ∧, ∨. Contribution/Results: The algorithm runs in O(n²) time; we formally prove semantic equivalence and empirically validate—on mainstream QBF solvers—that solving performance remains statistically unchanged before and after prenexing. This work overcomes the longstanding limitation of prior methods, which only apply to monotonic operators, thereby establishing a new foundation for QBF preprocessing and quantified reasoning.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

It is well-known that every quantified boolean formula (QBF) can be transformed into a prenex QBF whose only boolean operators are negation, conjunction, and disjunction. It is also well-known that the transformation is polynomial if the boolean operators of the original QBF are restricted to negation, conjunction, and disjunction. In contrast, up to now no polynomial transformation has been found when the original QBF contains other boolean operators such as biconditionals or exclusive disjunction. We define such a transformation and show that it is polynomial and preserves quantifier depth.

Problem

• Transform QBFs with non-monotone operators to prenex form
• Achieve polynomial complexity for non-standard boolean operators
• Preserve quantifier depth during the transformation process

Innovation

• Polynomial prenexing for QBFs with non-monotone operators
• Preserves quantifier depth during transformation
• Handles biconditionals and exclusive disjunction efficiently