A characterization of partial metrizability: domains are quantifiable

Abstract

A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Künzi in the survey paper Non-symmetric Topology (in: Proceedings of the Szekszard Conference, Bolyai Soc. Math. Studies, Vol. 4, 1993, pp. 303–338; problem 711“Characterize those quasi-uniformities having a countable base which are induced by a weighted quasi-metric”.). The characterization yields a powerful tool which establishes a correspondence between partial metrics and special types of valuations, referred to as Q-valuations (cf. also Theoret. Comput. Sci., to appear). The notion of a Q-valuation essentially combines the well-known notion of a valuation with a weaker version of the notion of a quasi-unimorphism, i.e. an isomorphism in the context of quasi-uniform spaces. As an application, we show that ω-continuous directed complete partial orders (dcpos) are quantifiable in the sense of O'Neill (in: S. Andima et al. (Eds.), Proceedings of the 11th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences, Vol. 86, 1997, pp. 304–315), i.e. the Scott topology and partial order are induced by a partial metric. For ω-algebraic dcpos the Lawson topology is induced by the associated metric. The partial metrization of general domains improves prior approaches in two ways: •The partial metric is guaranteed to capture the Scott topology as opposed to e.g. Smyth (Quasi-uniformities: Reconciling Domains with Metric Spaces, Lecture Notes in Computer Science, Vol. 298, Springer, Berlin, 1987, pp. 236–253), Bonsangue et al. (Theoret. Comput. Sci. 193 (1998) 1), Flagg (Theoret. Comput. Sci., to appear) and Flagg (Theoret. Comput. Sci. 177 (1) (1997) 1), which in general yield a coarser topology.•Partial metric spaces are Smyth-completable and hence their Smyth-completion reduces to the standard bicompletion. This type of simplification is advocated in Smyth (in: G.M. Reed, A.W. Roscoe, R.F. Wachter (Eds.), Topology and Category Theory in Computer Science, Oxford University Press, Oxford, 1991, pp. 207–229). Our results extend Smyth (1991)'s scope of application from the context of 2/3 SFP domains to general domains.The quantification of general domains solves an open problem on the partial metrizability of domains22Pawel Waszkiewicz communicated recently to the author that he obtained similar results independently. A stronger result implying the quantifiability of general domains is reported in [23]. O'Neill obtained the partial metrizability of ω-algebraic domains in his thesis [29]. stated in O'Neil (1997) and Heckmann (Appl. Categor. Struct. (1999) 71).Our proof of the quantifiability of domains is novel in that it relies on the central notion of a semivaluation (Schellekens, The correspondence between partial metrics and semivaluations, Theoret. Comput. Sci., to appear). The characterization of partial metrizability is entirely new and sheds light on the deeper connections between partial metrics and valuations commented on in [Bukatin and Shorina (in: M. Nivat (Ed.), Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, Vol. 1378, Springer, Berlin, 1998, pp. 125–139)]. Based on (Schellekens, The correspondence between partial metrics and semivaluations, Theoret. Comput. Sci., to appear) and our present characterization, we conclude that the notion of a (semi)valuation is central in the context of Quantitative Domain Theory since it can be shown to underlie the various models arising in the applications.