Products and projective limits of continuous valuations on <i>T</i><sub>0</sub> spaces

We give two concrete examples of continuous valuations on dcpo’s to separate minimal valuations, point-continuous valuations, and continuous valuations: (1) Let ${\mathcal J}$ be the Johnstone’s non-sober dcpo, and μ be the continuous valuation on ${\mathcal J}$ with μ(U)=1 for nonempty Scott opens U and μ(U)=0 for $U=\emptyset$ . Then, μ is a point-continuous valuation on ${\mathcal J}$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb{R}_\ell$ . Its restriction to the open subsets of $\mathbb{R}_\ell$ is a continuous valuation λ. Then, its image valuation $\overline\lambda$ through the embedding of $\mathbb{R}_\ell$ into its Smyth powerdomain $\mathcal{Q}\mathbb{R}_\ell$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $\overline\lambda$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo’s.

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A Domain-theoretic Approach to Statistical Programming Languages

Goubault-Larrecq,

Jia,

Théron

2023

J. ACM

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We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin’ s monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations. (Implementations by themselves are not addressed here.) Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: all measures on the real line can be modeled by minimal valuations on the domain $\mathbf {I}\mathbb {R}_\bot $ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples in order to demonstrate that our semantics computes exactly as one would expect, and in order to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance. Our examples also include some useful, non-trivial cases of distributions on higher-order objects.

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Products and projective limits of continuous valuations on T₀ spaces

Abstract: We show analogues of the Daniell–Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on T0 topological spaces.

Cited by 3 publications

References 21 publications

A few projective classes of (non-Hausdorff) topological spaces

A few projective classes of (non-Hausdorff) topological spaces

Separating minimal valuations, point-continuous valuations, and continuous valuations

A Domain-theoretic Approach to Statistical Programming Languages

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Products and projective limits of continuous valuations on T0 spaces

Abstract: We show analogues of the Daniell–Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on T0 topological spaces.

Cited by 3 publications

References 21 publications

A few projective classes of (non-Hausdorff) topological spaces

A few projective classes of (non-Hausdorff) topological spaces

Separating minimal valuations, point-continuous valuations, and continuous valuations

A Domain-theoretic Approach to Statistical Programming Languages

Contact Info

Product

Resources

About

Products and projective limits of continuous valuations on T₀ spaces