Abstract
The existence of deep connections between partial metrics and valuations is well known in domain theory. However, the treatment of non-algebraic continuous Scott domains has been not quite satisfactory so far.
In this paper we return to the continuous normalized valuations Μ on the systems of open sets and introduce notions of co-continuity ({U i , i ∃ I} is a filtered system of open sets ⟹ Μ(Int(∩ i∃I Ui)) = inf i∃I Μ(Ui)) and strong non-degeneracy (U ⊂ V are open sets ⟹ Μ(U) < Μ(V)) for such valuations. We call the resulting class of valuations CC-valuations. The first central result of this paper is a construction of CC-valuations for Scott topologies on all continuous dcpo's with countable bases. This is a surprising result because neither co-continuous, nor strongly non-degenerate valuations are usually possible for ordinary Hausdorff topologies.
Another central result is a new construction of partial metrics. Given a continuous Scott domain A and a CC-valuation Μ on the system of Scott open subsets of A, we construct a continuous partial metric on A yielding the Scott topology as u(x, y)=Μ(A (C x ∩ Cy)) − Μ(Ix ∩ Iy), where C x = {y ∃ A ¦ y ⊑ x } and I x = {y ∃ A ¦ {x, y } is unbounded }. This construction covers important cases based on the real line and allows to obtain an induced metric on Total(A) without the unpleasant restrictions known from earlier work.
Supported by Applied Continuity in Computations Project.
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Bukatin, M.A., Shorina, S.Y. (1998). Partial metrics and co-continuous valuations. In: Nivat, M. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 1998. Lecture Notes in Computer Science, vol 1378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0053546
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