A fundamental study into the theory and application of the partial metric spaces
O'Neill, Simon John
Our aim is to establish the partial metric spaces within the context of Theoretical Computer Science. We present a thesis in which the big "idea" is to develop a more (classically) analytic approach to problems in Computer Science. The partial metric spaces are the means by which we discuss our ideas. We build directly on the initial work of Matthews and Wadge in this area. Wadge introduced the notion of healthy programs corresponding to complete elements in a semantic domain, and of size being the extent to which a point is complete. To extend these concepts to a wider context, Matthews placed this work in a generalised metric framework. The resulting partial metric axioms are the starting point for our own research.\ud \ud In an original presentation, we show that Ta-metrics are either quasi-metrics, if we discard symmetry, or partial metrics, if we allow non-zero self-distances. These self-distances are how we capture Wadge's notion of size (or weight) in an abstract setting, and Edalat's computational models of metric spaces are examples of partial metric spaces. Our contributions to the theory of partial metric spaces include abstracting their essential topological characteristics to develop the hierarchical spaces, investigating their To-topological properties, and developing metric notions such as completions. We identify a quantitative domain to be a continuous domain with a To-metric inducing the Scott topology, and introduce the weighted spaces as a special class of partial metric spaces derived from an auxiliary weight function.\ud \ud Developing a new area of application, we model deterministic Petri nets as dynamical systems, which we analyse to prove liveness properties of the nets. Generalising to the framework of weighted spaces, we can develop model-independent analytic techniques. To develop a framework in which we can perform the more difficult analysis required for non-deterministic Petri nets, we identify the measure-theoretic aspects of partial metric spaces as fundamental, and use valuations as the link between weight functions and information measures. We are led to develop a notion of local sobriety, which itself appears to be of interest.
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