Recent developments in Domain Theory indicate that additional concepts are useful to develop the corresponding applications. These developments include domain theoretic approaches to dataflow networks, logic programming, domain theoretic approaches to integration, models for probabilistic languages and models for real number computation as well as models which incorporate complexity analysis.
Each of these applications involve “real number measurements” in some sense, and hence the adjective quantitative is used as opposed to the adjective qualitative which indicates the traditional order theoretic approach.
At this point several foundations exist. The more abstract approaches include the Yoneda completion, the continuity spaces and the topological quasi-uniform spaces. These approaches are essentially equivalent and lead to complex completions, involving non-idempotency or subtle relations between two topologies and a quasi-uniformity. Moreover, they involve generalized metrics which typically lead to topologies coarser than the Scott topology, which for instance for the case of topological quasi-uniform spaces is resolved by the addition of a new topology.
We introduce the basic tools involved in the theory and discuss some examples. We present a simplified approach to quantitative domains based on partial metrics for which the topological completion becomes standard. O'Neill and Heckmann raised the question which domains are “quantifiable” in the sense that there exists a partial metric which induces the Scott topology. O'Neill showed that Scott domains are quantifiable. We improve on this result by showing that ω-continuous dcpo's, and hence all domains, are quantifiable.