An introduction to metric semantics: operational and denotational models for programming and specification languages

Our focus is on the semantics of programming and speci$cation languages. Over the years, di1erent approaches to give semantics to these languages have been put forward. We restrict ourselves to the operational and the denotational approach, two main streams in the $eld of semantics. Two notions which play an important role in this paper are (non)determinism and (non)termination. Nondeterminism arises naturally in concurrent languages and it is a key concept in speci$cation languages. Nontermination is usually caused by recursive constructs which are crucial in programming. The operational models are based on labelled transition systems. The de$nition of these systems is guided by the structure of the language. Metric spaces are an essential ingredient of our denotational models. We exploit the metric structure to model recursive constructs and to de$ne operators on in$nite entities. Furthermore, we also employ the metric structure to relate operational and denotational models for a given language. On the basis of four toy languages, we develop some general theory for de$ning operational and denotational semantic models and for relating them. This theory is applicable to a wide variety of languages. We start with a very simple deterministic and terminating imperative programming language. By adding the recursive while statement, we obtain a deterministic and nonterminating language. Next, we augment the language with the parallel composition resulting in a bounded nondeterministic and nonterminating language. Finally, we add some timed constructs. We obtain an unbounded nondeterministic and nonterminating speci$cation language. c © 2001 Elsevier Science B.V. All rights reserved.