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

Goubault-Larrecq, Jean; Jia, Xiaodong

doi:10.1017/s0960129521000384

Cited by 3 publications

(1 citation statement)

References 19 publications

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“…Whether the Fubini-like formula above holds for every pair of continuous valuations µ and ν on arbitrary dcpos is unknown. We note that the problem would be easily solved if all continuous valuations were minimal, but that is not the case, as is shown in the paper [13].…”

Section: Monads Of Continuous R-valuationsmentioning

confidence: 92%

Continuous R-valuations

Jean¹,

Jia²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

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Section: Monads Of Continuous R-valuationsmentioning

confidence: 92%

Continuous R-valuations

Jean¹,

Jia²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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A construction of free dcpo-cones

Chen,

Kou,

Lyu

et al. 2024

Math. Struct. Comp. Sci.

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We give a construction of the free dcpo-cone over any dcpo. There are two steps for getting this result. Firstly, we extend the notion of power domain to directed spaces which are equivalent to $T_0$ monotone-determined spaces introduced by Erné, and we construct the probabilistic powerspace of the monotone determined space, which is defined as a free monotone determined cone. Secondly, we take D-completion of the free monotone determined cone over the dcpo with its Scott topology. In addition, we show that generally the valuation power domain of any dcpo is not the free dcpo-cone.

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

Cited by 3 publications

References 19 publications

Continuous R-valuations

Continuous R-valuations

A construction of free dcpo-cones

A Domain-theoretic Approach to Statistical Programming Languages

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