We introduce an enhanced notion of unsatisfiable cores for QBF in prenex CNF that allows to weaken universal quantifiers to existential quantifiers in addition to the traditional removal of clauses. The resulting unsatisfiable cores can be different from those of the traditional notion in terms of syntax, standard semantics, and proof-based semantics. This not only gives rise to explanations of unsatisfiability but, via duality, also leads to diagnoses and repairs of unsatisfiability that are not obtained with traditional unsatisfiable cores. We use a source-to-source transformation on QBF in PCNF such that the weakening of universal quantifiers to existential quantifiers in the original formula corresponds to the removal of clauses in the transformed formula. This makes any tool or method for the computation of unsatisfiable cores of the traditional notion available for the computation of unsatisfiable cores of our enhanced notion. We implement our approach as an extension to the QBF solver DepQBF, and we perform an extensive experimental evaluation on a subset of QBFLIB. We illustrate with several case studies that helpful information can be provided by unsatisfiable cores of our enhanced notion.

中文翻译：

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QBF 的增强不可满足核心：弱化通用到存在量词

我们在 prenex CNF 中引入了 QBF 不可满足核心的增强概念，除了传统的删除子句之外，它还允许将全称量词弱化为存在量词。由此产生的不可满足核心在语法、标准语义和基于证明的语义方面可能与传统概念不同。这不仅引起了对不可满足性的解释，而且通过对偶性，还导致了传统不可满足核心无法获得的不可满足性的诊断和修复。我们在 PCNF 中的 QBF 上使用源到源转换，使得原始公式中的全称量词弱化为存在量词对应于转换后的公式中子句的删除。这使得任何用于计算传统概念的不可满足核心的工具或方法都可用于计算我们增强概念的不可满足核心。我们将我们的方法作为 QBF 求解器 DepQBF 的扩展来实现，并且我们对 QBFLIB 的一个子集进行了广泛的实验评估。我们用几个案例研究来说明有用的信息可以由我们增强概念的不可满足的核心提供。