Applications of compactness in the Smyth powerdomain of streams [chapter]

J. -J. Ch. Meyer, E. P. Vink
1987 Lecture Notes in Computer Science  
We show in a uniform setting the crucial role of compactness in the theory of the Smyth powerdomain of streams. The topological notion of compactness is characterized in an order-theoretical manner, involving a notion of bounded sets. We obtain general results on the continuity of operators, and consider applications as diverse as interleaving, hiding and stream programming operators. Section 1 Introduction This paper originated from writing a tutorial for students on (denotational) stream
more » ... tics of concurrency, such as used by e.g. Broy ([Brl], [Br2]), Back ([BAD and De Bakker et ai. ([BKMOZ]). in the setting of this tutorial programs are built from atomic actions, sequential composition, alternative composition, parallel composition (interleaving or merge) and recursive constructs. In [Mel], [Me2] it was studied how to develop a uniform semantics for such a language based upon a powerdomain ordered by the Smyth ordering (of. [Sml]), which appeared to be equivalent to an observation-based semantics in the style of Hoare et al., (of. [OH], [BMO]). (A recent application of the Smyth powerdomain to the semantics of more refined concurrency languages can be found in [Ma].) In the framework of [Mel], [Me2] atomic actions are left uninterpreted, so denotations of program statements are sets of streams (traces) of atomic actions. Moreover, it is assumed that the set of atomic actions is finite. Some of the proofs, however, are difficult and not suited for lecture notes for students. In the meantime we found that using the Egli-Milner order instead of the Smyth ordering provided a simplification of the proofs. So when we wrote the tutorial we chose for a stream powerdomain ordered by the Egli-Milner ordering, and as such the material could be viewed as an extension of Back's work [Ba] for concurrent operators such as interleaving. During the elaboration of the material for the lecture notes we discovered that regarding the continuity proofs of the operators, viz. sequential, alternative and parallel composition, the condition of the finiteness of the alphabet of atomic actions could be relaxed to a certain condition of boundedness of the stream-sets that occur as denotations. Informally, a stream-set is bounded if all sets of truncations up to a finite length are finite. The continuity proofs now became very elegant and uniform. After the completion of the lecture notes we felt that it was interesting enough to write an article on the subject. At some moment we realized that the concept of boundedness could also be employed profitably in the case of the Smyth powerdomain. Moreover, we learned that in this case it was even more crucial. Whereas in the study of the Egli-Milner domain the boundedness concept proved to be a useful tool to ease the continuity proofs, in the Smyth powerdomain it is an essential requirement for the continuity of the
doi:10.1007/3-540-17660-8_59 fatcat:ir4f556pvvddpbchro7vhffjtq