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Admissible representations of probability measures
 Electronic Notes in Theoretical Computer Science
"... In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We ..."
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In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. This canonical representation turns out to be admissible with respect to the weak topology on M(X). Moreover, we prove that for countably based topological spaces X the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology on M(X).
Representing Probability Measures using Probabilistic Processes
 Journal of Complexity
, 2006
"... In the Type2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the repres ..."
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In the Type2 Theory of Effectivity, one considers representations of topological spaces in which infinite words are used as “names ” for the elements they represent. Given such a representation, we show that probabilistic processes on infinite words generate Borel probability measures on the represented space. Conversely, for several wellbehaved types of space, every Borel probability measure is represented by a corresponding probabilistic process. Accordingly, we consider probabilistic processes as providing “probabilistic names ” for Borel probability measures. We show that integration is computable with respect to the induced representation of measures. 1
Two Preservation Results for Countable Products of Sequential Spaces
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
"... We prove two results about the sequential topology on countable products of sequential topological spaces. First, we show that a countable product of topological quotients yields a quotient map between the product spaces. Second, we show that the reflection from sequential spaces to its subcategory ..."
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We prove two results about the sequential topology on countable products of sequential topological spaces. First, we show that a countable product of topological quotients yields a quotient map between the product spaces. Second, we show that the reflection from sequential spaces to its subcategory of monotone ωconvergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on nondiscrete spaces.
An Effective TietzeUrysohn Theorem for QCBSpaces
 Journal of Universal Computer Science, Vol
, 2009
"... Abstract: The TietzeUrysohn Theorem states that every continuous realvalued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we in ..."
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Abstract: The TietzeUrysohn Theorem states that every continuous realvalued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we introduce for qcbspaces a slightly weaker notion of normality than the classical one and show that this property suffices to establish an Extension Theorem for continuous functions defined on functionally closed subspaces. Qcbspaces are known to form an important subcategory of the category Top of topological spaces. QCB is cartesian closed in contrast to Top.
Computing with Sequences, Weak Topologies and the Axiom of Choice
 Proc. Computer Science Logic 2005. Springer LNCS 3634
"... Abstract. We study computability on sequence spaces, as they are used in functional analysis. It is known that nonseparable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice nonseparable normed spaces cannot even be admissibly represented wi ..."
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Abstract. We study computability on sequence spaces, as they are used in functional analysis. It is known that nonseparable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice nonseparable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology is one which makes all bounded linear functionals continuous). Surprisingly, it turns out that when one replaces the Axiom of Choice by the Axiom of Dependent Choice and the Baire Property, then some nonseparable normed spaces can be represented admissibly on Turing machines with respect to the weak topology (which is just the weakest compatible topology). Thus the ability to adequately handle sequence spaces on Turing machines sensitively relies on the underlying axiomatic setting. 1
Comparing free algebras in Topological and Classical Domain Theory. Submitted, 2006. (Available from http://homepages.inf.ed.ac.uk/als/Research/topologicaldomaintheory.html) [6
 Math. Struct. of Comp. Science
, 2006
"... We compare how computational effects are modelled in Classical Domain Theory and Topological Domain Theory. Both of these theories provide powerful toolkits for denotational semantics: Classical Domain Theory being introduced by Scott, and wellestablished and developed since; Topological Domain Th ..."
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We compare how computational effects are modelled in Classical Domain Theory and Topological Domain Theory. Both of these theories provide powerful toolkits for denotational semantics: Classical Domain Theory being introduced by Scott, and wellestablished and developed since; Topological Domain Theory being a generalization in which topologies more general than the Scotttopology are admitted. Computational effects can be modelled using free algebra constructions, according to Plotkin and Power, and we show that for a wide range of computational effects, including all the classical powerdomains, this free algebra construction coincides in Classical and Topological Domain Theory, when restricted to countablybased continuous domains. 1
Additional services for Mathematical Structures in
, 2009
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