An operational and a denotational semantic model are presented for a real-time programming language incorporating the concept of integration. Integration enables us to specify a restricted form of unbounded non-determinism. For example, the execution of an action at an arbitrary moment in a time interval can be specified using the concept of integration. The operational and the denotational model are proved to be equivalent using a general method based on higher-order transformations and complete metric spaces. Banach~s fixed point theorem, which states that a contraction on a complete metric space has a unique fixed point, will be used to define and to compare semantic models. Also Michael's theorem, which roughly states that a compact union of compact sets is compact, will be used for the definition of semantic models.
I n t r o d u c t i o nI~eal-time programming can be viewed as traditional concurrency supplied with timing constraints [81]. Because these timing constraints cause more complexity, the advantages of high-level languages are even greater in real-time programming than in concurrency and sequential programming. Several languages, like RTL [22], have been designed specifically for real-time programming. Other languages are extensions of already existing languages, for example, the language TCSP [14] is an extension of the language CSP [21]. In real-time programming the correctness of a program depends not only on the flow of control. The program should also meet its timing constraints [30]. Therefore new semantic models should be developed. Several models both operational [2,16,19] and denotational [15,24,27] have already been provided.In this paper a simple real-time programming language is studied. Apart from the traditional programming constructs, this language incorporates timed atomic actions and integration. Timed atomic actions are atomic actions each provided with some timing information. This timing information denotes when the atomic action should be executed. The concept of integration has been introduced in [2]. A statement is integrated over a time set, which is a subset of the time domain, i.e. a non-deterministically chosen value from the time set is passed to the statement. Integration enables us to specify the execution of an action at an arbitrary moment in a time interval, for example, fte [1.o3, 2.tq (a,t)