# Solving Dependency Quantified Boolean Formulas Using Quantifier Localization

## TL;DR

This paper introduces a formal framework for non-prenex DQBFs, enabling quantifier localization, and demonstrates its effectiveness by enhancing a state-of-the-art solver with experimental validation on benchmark instances.

## Contribution

It provides the first formal definition and properties for non-prenex DQBFs and integrates quantifier localization into existing solvers.

## Key findings

- Quantifier localization improves solver performance.
- The approach is effective on benchmark instances.
- Non-prenex DQBFs can be formally defined and utilized.

## Abstract

Dependency quantified Boolean formulas (DQBFs) are a powerful formalism, which subsumes quantified Boolean formulas (QBFs) and allows an explicit specification of dependencies of existential variables on universal variables. Driven by the needs of various applications which can be encoded by DQBFs in a natural, compact, and elegant way, research on DQBF solving has emerged in the past few years. However, research focused on closed DQBFs in prenex form (where all quantifiers are placed in front of a propositional formula), while non-prenex DQBFs have almost not been studied in the literature. In this paper, we provide a formal definition for syntax and semantics of non-closed non-prenex DQBFs and prove useful properties enabling quantifier localization. Moreover, we make use of our theory by integrating quantifier localization into a state-of-the-art DQBF solver. Experiments with prenex DQBF benchmarks, including all instances from the QBFEVAL'18-'20 competitions, clearly show that quantifier localization pays off in this context.