Complexity Spaces Revisited

The complexity (quasi-pseudo-metric) spaces have been introduced as part of the development of a topological foundation for the complexity analysis of algorithms ((Sch95]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in Sch95]. Typically these applications involve xed point arguments based on the Smyth completion. The notion of S-completability plays an important role in the study of the complexity spaces since it implies a simpliication of the Smyth completion to the bicompletion (e.g. Sch95], Smy92] and S un91]). A characterization of S-completable quasi-uniform spaces has been given in S un91]. We present a related characterization of S-completable quasi-pseudo-metric spaces and provide a simpliied proof of the fact that weightable quasi-pseudo-metric spaces are S-completable ((K un93]). We recall that the weightable quasi-pseudo-metric spaces and the totally bounded quasi-pseudo-metric spaces ((Smy91] and S un91]) form the two main examples of classes of S-completable quasi-pseudo-metric spaces encountered in the literature. Complexity spaces are S-completable since they form a class of weightable spaces ((Sch95]). Complexity spaces with a complexity lower bound are shown to be totally bounded and this result is discussed in the light of Smyth's computational interpretation of totally boundedness ((Smy91]).