L-domains as locally continuous sequent calculi

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.