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The probabilistic powerdomain for stably compact spaces

Mauricio Alvarez-Manilla, Achim Jung, Klaus Keimel
2004 Theoretical Computer Science  
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

Klaus Keimel
2004 Electronical Notes in Theoretical Computer Science  
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]

Ben Cohen, Martin Escardo, Klaus Keimel
2006 Lecture Notes in Computer Science  
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

Matthias Schröder, Alex Simpson
2006 Electronical Notes in Theoretical Computer Science  
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

Jean Goubault-Larrecq
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
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]

M. Andrew Moshier, Achim Jung
2002 Lecture Notes in Computer Science  
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

JIMMIE LAWSON
2010 Mathematical Structures in Computer Science  
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

Jean Goubault-Larrecq, Martín Escardó
2012 Logical Methods in Computer Science  
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]

Achim Jung
2013 Lecture Notes in Computer Science  
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

Regina Tix, Klaus Keimel, Gordon Plotkin
2005 Electronical Notes in Theoretical Computer Science  
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

Regina Tix, Klaus Keimel, Gordon Plotkin
2009 Electronical Notes in Theoretical Computer Science  
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

Klaus Keimel
2006 Electronical Notes in Theoretical Computer Science  
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

Klaus Keimel
2008 Topology and its Applications  
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

Jean Goubault-Larrecq, Achim Jung
2014 Electronical Notes in Theoretical Computer Science  
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]

Jean Goubault-Larrecq
2020 arXiv   pre-print
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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