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-com...

A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Künzi in the survey paper Nonsymmetric Topology ([Kün93], problem 71). 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 [Sch00]). The notion of a Q-valuatio...

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric: it can be thought of as an asymmetric metric. Quasi-metrics were first introduced in 1930s and are a subject of intensive research in the context of topology and theoretical computer science. The central result of this thesis, developed in Chapter 3, is that a natural correspondence exists betwee...

If the symmetry condition in the definition of a pseudometric is deleted, the notion of a quasi-pseudometric is obtained. Asymmetric distance functions already occur in the work of Hausdorff in the beginning of the twentieth century when in his book on set-theory [27] he discusses what is now called the Hausdorff metric of a metric space. A family of pseudo-metrics on a set generates a uniformi...