Applications -- what are the known or conceived uses for partial metric spaces?Presently there are two suggested uses for partial metric spaces. On this page we first discuss how partial metric spaces are employed in Computer Science, and after that, what they contribute to the foundation of Mathematics.Partial metric spaces and Computer ScienceThe known and conceived uses in computer science are to be found in the application of domain theory. It is recognised that domain theory may need additional machinery in order to model specific situations. Where this would require the correct modelling of real numbers for perhaps measurement or arithmetic, a concept of quantity is introduced to Scott's theory of domains, which is, by way of contrast, termed qualitative. The term quantitative domain theory has thus been coined for any study which sets out to (correctly) combine real numbers with domains.Analysis of data flow deadlockPartial metric spaces arose not as a general theory to unify domains with real numbers. Rather, they resulted from one specific domain requiring a metric-style analysis in order to prove absence of deadlock in a class of data flow networks [Wad81, Mat95]. As such partial metric spaces never became a general theory of quantitative domains. Unifications of domain theory with partial metric spaces seem to require the generalisation of the real numbers to quantales [KMP04], an analogous result holding for unifying domains with metric spaces [Wag94].Complexity analysis of programsBesides data flow, there are other interesting specific examples of useful domains which are pmetrizable. Thus although not a general theory of quantitative domains, partial metric spaces do have a role to play in the application of domain theory where the domain happens to be weightable. Schellekens has used partial metric spaces to analyse semantically the complexity of programs using divide and conquer algorithms [Sch95].Corect modelling of exact real number arithmeticAchim Jung has suggested (privately) that partial metric spaces would be useful for reasoning about convergence properties in Martin Escardo's work on the exact modelling of real number arithemtic in domain theory.FractalsKim Wagner has suggested [Wag94] that the modelling of fractals would be a good example of where domain theory and metric spaces need to work together; would such a framework be pmetrizable?Partial metric spaces and MathematicsIt is known that any topology can arise from a generalised metric [Ko88]. A later result says that any topology can arise from a partial metric [KMP04], although the range of non negative reals will need to be generalised to a quantale. Consider this result in the context where (correct) computable models of many branches of real valued analysis are being developed. This involves classical analysis, non symmetric topology, domain theory, and programming language design all coming together into a new mathematics. Partial metric spaces are a candidate for describing mathematics that has to be simultaneously understood as being qualitative and quantitative. |