Continuous Denotational Semantics

Abstract

The development of a programming logic in the scope of classical first -order logic is connected with making the formal notions, used to describe program properties, internal. This can be ensured by defining the definability of these notions in the logic in question. Here we show how dynamic logic can be developed in classical first-order logic by internalizing the denotation Den only. This internalization can be achieved by an appropriate axiomatization of reflexive and transitive closure. For the latter we introduce in Section 14.1 the so-called transitive extension and the corresponding extended language. This language will be appropriate to introduce two axiom systems Ind ^σ and Ind ^*_σ . In this section it will also be shown how these axiom systems can be used to define the denotational semantics. In Section 14.2, by using the axiom systems Ind ^σ and Ind ^*_σ we introduce the dynamic logic with continuous denotational semantics DL ^ns_σ and that with strongly continuous denotational semantics DL ^ns*_σ . The first one we also call non-standard dynamic logic. These logics realize our aim to define dynamic logic in the scope of classical first-order logic. Moreover, they preserve the classical notion of completeness, and we will show that they are compact and complete. Considering the Hoare calculus and its generalization we will show that these calculi are complete with respect to the continuous and the strongly continuous denotational semantics, respectively. These results at the same time provide a first-order characterization of the Hoare calculus and of its generalization.