Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent L-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an L-domain, and every L-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of L-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting.

受多语言序贯演算框架的启发，我们引入了一种称为局部连续序贯演算的形式逻辑系统来表示L域。通过考虑局部连续后继演算上定义的逻辑状态，我们证明了局部连续后继演算的所有逻辑状态关于集合包含的集合形成了一个L域，并且每个L域都可以通过这种方式获得。此外，我们将合取结果关系定义为我们的连续演算之间的态射，并证明局部连续连续演算和合取结果关系的范畴等价于L域和斯科特连续函数的范畴。这一结果将阿布拉姆斯基的“逻辑形式的域理论”扩展到了连续的环境。