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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

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  ...  Via this identification the duality in Theorem 8.21 can be restated as a duality between the de Groot dual of the probabilistic powerdomain on Q(X) and the probabilistic powerdomain on the Hoare powerdomain  ... 
doi:10.1017/s0960129510000319 fatcat:uwsasrcwgvasfgr6ruvuhj7squ

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

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

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

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).  ...  functionals on continuous directed complete partially ordered cones, endowed with their Scott topologies, in analogy to the corresponding theorems for topological vector spaces in functional analysis  ...  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

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 form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders.  ...  We characterise the algebras of the probabilistic powerdomain monad over compact ordered spaces: These are the compact ordered convex spaces embeddable in locally convex ordered topological vector spaces  ... 
doi:10.1016/j.topol.2008.07.002 fatcat:cwvmsbcnk5eule6ti2jhydnw3i

Stably Compact Spaces and the Probabilistic Powerspace construction

Achim Jung
2004 Electronical Notes in Theoretical Computer Science  
We then exhibit some constructions for stably compact spaces which apparently have not appeared in the literature before.  ...  We put forward a revised definition of stably compact spaces which allows us to show their equivalence with Nachbin's compact ordered spaces in an entirely elementary fashion.  ...  Finally, in [JT98] we showed that the probabilistic powerdomain of a coherent domain is again coherent.  ... 
doi:10.1016/j.entcs.2004.10.001 fatcat:nwdxvlnftnd3ner27ru5kb2wvy

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.  ...  This paper reviews the one-to-one correspondence between stably compact spaces (a topological concept covering most classes of semantic domains) and compact ordered Hausdorff spaces.  ...  Compact ordered and stably compact spaces Compact ordered spaces A partially ordered topological space (or ordered space, for short) in the sense of Nachbin [29] is a set X with a topology O and a  ... 
doi:10.1016/j.tcs.2004.06.021 fatcat:yrnxbcieszbqzcuct65226nina

On the computational content of the Lawson topology

Frédéric De Jaeger, Martín Escardó, Gabriele Santini
2006 Theoretical Computer Science  
Among other examples, we consider the upper powerdomain of Euclidean space, for which we prove that Scott and Lawson computability coincide with two notions of computability for compact sets recently proposed  ...  For example, in the powerset of the natural numbers with its standard effective presentation, an element is Scott computable iff it is a recursively enumerable set, and it is Lawson computable iff it is  ...  He is grateful to Giuseppe Longo for the invitation, and to all members of the DI for the hospitality. The authors are also grateful to Giuseppe Longo for lively technical discussions.  ... 
doi:10.1016/j.tcs.2006.03.021 fatcat:zucrpn3ukbek7nn3irgxpfxrwi

Presenting de Groot duality of stably compact spaces [article]

Tatsuji Kawai
2020 arXiv   pre-print
We give a constructive account of the de Groot duality of stably compact spaces in the setting of strong proximity lattice, a point-free representation of a stably compact space.  ...  The new notion allows us to identify de Groot duals of stably compact spaces by analysing the duals of their presentations.  ...  I thank the institute for their support and the organisers of the program for creating a stimulating environment for research.  ... 
arXiv:1812.06480v4 fatcat:pzvopsqxmjhhffbdlwkuazhmfm

Topological cones: functional analysis in a T0-setting

Klaus Keimel
2008 Semigroup Forum  
The theme of topological cones has been reappearing in the author's work in a non-Hausdorff setting motivated by the needs of mathematical models for a denotational semantics of languages combining probabilistic  ...  Already in his PhD Thesis on compact Abelian semigropups under the direction of Karl Heinrich Hofmann the author was lead to investigate locally compact cones [18] .  ...  This applies in particular to the the extended probabilistic powerdomain V(X) over a stably locally compact space.  ... 
doi:10.1007/s00233-008-9078-0 fatcat:hvsfuersyvbknnx4ll4ruoigfa

Stably Compact Spaces and Closed Relations

Achim Jung, Mathias Kegelmann, M.Andrew Moshier
2001 Electronical Notes in Theoretical Computer Science  
Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints.  ...  In this paper we let the morphisms between stably compact spaces be certain "closed relations" and study the resulting categorical properties.  ...  Acknowledgements The authors are grateful for the many comments they received when parts of this research were presented at earlier occasions.  ... 
doi:10.1016/s1571-0661(04)80964-2 fatcat:m4ipxt2za5dcdnhxjed73xb3uy

Continuity spaces: Reconciling domains and metric spaces

Bob Flagg, Ralph Kopperman
1997 Theoretical Computer Science  
the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming.  ...  Our proposal for a general notion of semantic domain is a symmetrically compact Y-continuity space, where V is a value quantale.  ...  the manuscript and Jimmie Lawson and Horst Herrlich for helping us understand the issue of Cartesian closedness for categories of <domains.  ... 
doi:10.1016/s0304-3975(97)00236-3 fatcat:qm4kami47zdvzoalvb4germkpu
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