Continuous Domains and their Information System Representation as Logical Systems

Abstract

In this work, we will introduce a finite information logic for continuous domains, as a complementary logic to continuous domain logic [BA95,Bed96,BA99,Keg99]. Here, the logical aspects of a continuous domain are analysed instead of the whole class. In order to do so we will use the continuous information systems representation of continuous domains [Hoo93,Bed96] and we relate both structures via an isomorphism. This triple (continuous domain, isomorphism and continuous information system) are called continuous logical systems. From a continuous logical system we can define several notions, such as: satisfaction relation, information content, model, theory, logical consequence, information content order, etc. Based on the constructors of continuous domains and continuous information systems, we will define the respectives constructors of continuous logical systems. Finally, we will extract a continuous domain logic from continuous logical systems and provide a soundness and completeness result for this logic based on the notion of model.