Focusing and selection are techniques that shrink the proof-search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atomic formula can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory and the related framework called superdeduction; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof-search space.

中文翻译：

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将聚焦和分辨率与选择联系起来

聚焦和选择是分别缩小后续演算和分辨率的证明搜索空间的技术。为了找出它们之间的联系，我们对它们都进行了概括：我们引入了一个连续的微积分，其中每个发生原子式可以有正极性或负极性；以及一种解析方法，其中可以在输入子句中选择每个文字，无论其符号如何。我们证明了该连续演算中的无截证明与该解决方法中空子句的推导之间的等价性。这种概括通常不是半完整的，它允许我们考虑与任何逻辑强度的理论相对应的完整实例。我们提出了三个完整的例子：首先，我们的框架允许我们证明普通聚焦对应于超分辨率和语义分辨率；第二个例子是演绎模理论及其相关框架超演绎；一个新的设置，没有被任何现有框架捕获，扩展了模数推导理论，重写了具有多个左侧的规则，