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The probabilistic powerdomain for stably compact spaces
2004
Theoretical Computer Science
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If these are restricted to the probabilistic or sub-probabilistic case, then another stably compact space is obtained. ...
As an application of this, the Riesz Representation Theorem is used for a straightforward proof of the (known) fact that every valuation on a stably compact space extends uniquely to a Radon measure on ...
The natural setting for our results, then, is that of stably compact spaces. ...
doi:10.1016/j.tcs.2004.06.021
fatcat:yrnxbcieszbqzcuct65226nina
The Probabilistic Powerdomain for Stably Compact Spaces via Compact Ordered Spaces
2004
Electronical Notes in Theoretical Computer Science
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In this paper the probabilistic powerdomain of a stably compact space Xs is investigated using the compact ordered space X and classical tools from measure theory and functional analysis. ...
Stably compact spaces Xs can be described as being derived from compact ordered spaces X by weakening their topology to the open upper sets. ...
Part (5) of the Theorem implies that, for a stably compact space X, the probabilistic powerdomains V ≤1 (X) and V 1 (X) are again stably compact. This has also been proved by Alvarez Manilla [2] . ...
doi:10.1016/j.entcs.2004.10.002
fatcat:nhyitjszyrfotiew62mxlnsnpq
The Extended Probabilistic Powerdomain Monad over Stably Compact Spaces
[chapter]
2006
Lecture Notes in Computer Science
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It is the aim of this work to obtain similar results for the (extended) probabilistic powerdomain monad over stably compact spaces. ...
For the semantics of probabilistic features in programming mainly two approaches are used for building models. ...
Let (A, α) be an Eilenberg-Moore algebra of the extended probabilistic powerdomain monad V over the category of stably compact spaces. Then A is a stably compact topological cone. ...
doi:10.1007/11750321_54
fatcat:s3gjcsnqifgddic3g2e4d2kwsa
Probabilistic Observations and Valuations
2006
Electronical Notes in Theoretical Computer Science
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We give a universal property for an "abstract probabilistic powerdomain" based on an analysis of observable properties of probabilistic computation. ...
In the category of topological spaces, the abstract probabilsitic powerdomain is given explicitly as the space of continuous probability valuations with weak topology. ...
Corollary 4. 4 4 The abstract probabilistic powerdomain over a stably compact space X is homeomorphic to the space M 1 (X) of regular Borel probability measures with the weak topology. ...
doi:10.1016/j.entcs.2005.11.075
fatcat:5f2a7oafhvhshpzbrwplqjzt64
omega-QRB-Domains and the Probabilistic Powerdomain
2010
2010 25th Annual IEEE Symposium on Logic in Computer Science
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This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. ...
We show that they form a category QRB with pleasing properties: QRB is closed under the probabilistic powerdomain functor, has all finite products, all bilimits, and is stable under retracts, and even ...
The category of qcb-spaces accommodates two probabilistic powerdomains [5] . ...
doi:10.1109/lics.2010.50
dblp:conf/lics/Goubault-Larrecq10
fatcat:xplyvmen5jhrbie7gkm2kjyr4m
A Logic for Probabilities in Semantics
[chapter]
2002
Lecture Notes in Computer Science
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We discuss the problem of integrating probabilistic and non-deterministic computation after presenting the construction of logics for probabilistic powerdomains. ...
powerdomains that arise directly from given logics for the underlying domains. ...
Acknowledgements The research reported here was started when the first author visited the School of Computer Science of the University of Birmingham in the Summer of 2001, supported by a guest professorship ...
doi:10.1007/3-540-45793-3_15
fatcat:j2jzttl65be2vhr4m5zi4c6qy4
Stably compact spaces
2010
Mathematical Structures in Computer Science
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inner and outer pavings is a useful and natural tool for such studies of stably compact spaces. ...
The purpose of this article is to develop the basic theory of stably compact spaces (=compact, locally compact, coherent sober spaces) and introduce in an accessible manner with a minimum of prerequisites ...
We outline how we use the earlier machinery of this paper to establish that the probabilistic powerdomain of a stably compact space is again stably compact. ...
doi:10.1017/s0960129510000319
fatcat:uwsasrcwgvasfgr6ruvuhj7squ
QRB-Domains and the Probabilistic Powerdomain
2012
Logical Methods in Computer Science
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This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. ...
We show that they form a category omega-QRB with pleasing properties: omega-QRB is closed under the probabilistic powerdomain functor, under finite products, under taking bilimits of expanding sequences ...
Plotkin for discussions, the anonymous referees at LICS'2010, and the anonymous referees of this paper, who offered counterexamples to conjectures, and numerous simplifications and improvements in almost ...
doi:10.2168/lmcs-8(1:14)2012
fatcat:df7czzs24bhhzorqtckcavyxmu
Continuous Domain Theory in Logical Form
[chapter]
2013
Lecture Notes in Computer Science
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As a first approximation, in a stably compact space the compact saturated sets behave exactly as compact sets do in Hausdorff spaces. ...
The definition of stably compact space is a bit involved and the interested reader is referred to [Jun04] or [GHK + 03] for a precise definition. ...
doi:10.1007/978-3-642-38164-5_12
fatcat:bnsmbo3n5zckvhj5znprxwcxcq
RETRACTED: Semantic Domains for Combining Probability and Non-Determinism
2005
Electronical Notes in Theoretical Computer Science
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Extending slightly a recent result from [3], we now know that the extended probabilistic powerdomain is Lawson-compact over any stably locally compact space. ...
probabilistic powerdomain over a Lawson-compact continuous domain is Lawsoncompact. ...
As a (continuous) retract of the stably compact space MV(X), the space V(X) with the weak topology is stably compact (see e.g. [3, Proposition 16] ). (d) Let X be a continuous domain. ...
doi:10.1016/j.entcs.2004.06.063
fatcat:kcsyh7rsgncadlrlnlonoaraau
Semantic Domains for Combining Probability and Non-Determinism
2009
Electronical Notes in Theoretical Computer Science
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The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4). ...
Our construction combines the well-known domains modelling nondeterminism -the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [24] modelling probabilistic ...
As a (continuous) retract of the stably compact space MV(X), the space V(X) with the weak topology is stably compact (see e.g. [3, Proposition 16] ). (d) Let X be a continuous domain. ...
doi:10.1016/j.entcs.2009.01.002
fatcat:d25ftdfq6rbv7ivv6f7olglarq
Topological Cones: Foundations for a Domain Theoretical Semantics Combining Probability and Nondeterminism
2006
Electronical Notes in Theoretical Computer Science
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In this extended abstract we intend to show that the theory can be extended to larger classes of spaces including in particular stably locally compact state spaces. Proofs are mostly omitted. ...
A motivating situation was the semantics of an imperative language with both probabilistic and nondeterministic choice as considered by McIver and Morgan [9,10] for discrete state spaces. ...
These new powerdomain construction can be applied in particular to the probabilistic powerdomain V(X) over a locally stably compact space X. ...
doi:10.1016/j.entcs.2005.11.066
fatcat:m73opjnzajc75jcdunbkkf7ku4
The monad of probability measures over compact ordered spaces and its Eilenberg–Moore algebras
2008
Topology and its Applications
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This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [ ...
The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. ...
One would hope to be able to combine the results known from both sides for a unifying characterisation of the algebras of the probabilistic powerdomain monad over stably compact spaces, thus reconciling ...
doi:10.1016/j.topol.2008.07.002
fatcat:cwvmsbcnk5eule6ti2jhydnw3i
QRB, QFS, and the Probabilistic Powerdomain
2014
Electronical Notes in Theoretical Computer Science
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The equivalence with sober QFS-spaces is then applied to give a novel, direct proof that the probabilistic powerdomain of a QRB-domain is a QRB-domain. ...
We show that the first author's QRB-domains coincide with Li and Xu's QFS-domains, and also with Lawson-compact quasi-continuous dcpos, with stably-compact locally finitary compact spaces, with sober QFS-spaces ...
over QRB-Domains Let us turn to the probabilistic powerdomain V(X) over a space X. ...
doi:10.1016/j.entcs.2014.10.010
fatcat:uvgt3o2hjbd6dp5niusihnx66y
Probabilistic Powerdomains and Quasi-Continuous Domains
[article]
2020
arXiv
pre-print
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The probabilistic powerdomain V X on a space X is the space of all continuous valuations on X. ...
We also show that the Scott and weak topologies on the V X may differ when X is not quasi-continuous, and we give a simple, compact Hausdorff counterexample. ...
The space of all continuous valuations on a topological space X is called the probabilistic powerdomain VX on X. ...
arXiv:2007.04189v3
fatcat:4ut4g57awfgvvjmoazxdi6rusy
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