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The monad of probability measures over compact ordered spaces and its EilenbergMoore algebras
, 2008
"... The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map f: K → L of compact Hausdorff spaces induces a continuous affine map Pf: PK → PL extending P. Together with the canonical embedding ε: K → PK ..."
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The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map f: K → L of compact Hausdorff spaces induces a continuous affine map Pf: PK → PL extending P. Together with the canonical embedding ε: K → PK associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P, ε, β). The EilenbergMoore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [31]. We generalise this result to compact ordered spaces in the sense of Nachbin [23]. 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. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces. This result can be seen as a step towards the characterisation of the algebras of the monad of probability
The extended probabilistic power domain monad over stably compact spaces. In
 Li (Eds.), Proceedings of TAMC06: Theory and Applications of Models of Computation, Bejing 2006. Lecture Notes in Computer Science 3959 (2006
, 1978
"... Abstract. For the semantics of probabilistic features in programming mainly two approaches are used for building models. One is the Giry monad of Borel probability measures over metric spaces, and the other is Jones ’ probabilistic powerdomain monad [6] over dcpos (directed complete partial orders). ..."
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Abstract. For the semantics of probabilistic features in programming mainly two approaches are used for building models. One is the Giry monad of Borel probability measures over metric spaces, and the other is Jones ’ probabilistic powerdomain monad [6] over dcpos (directed complete partial orders). This paper places itself in the second domain theoretical tradition. The probabilistic powerdomain monad is well understood over continuous domains. In this case the algebras of the monad can be described by an equational theory [6, 9, 5]. It is the aim of this work to obtain similar results for the (extended) probabilistic powerdomain monad over stably compact spaces. We mainly want to determine the algebras of this powerdomain monad and the algebra homomorphisms. 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
On the equivalence of state transformer semantics and predicate transformer semantics
, 2012
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On CSP and the Algebraic Theory of Effects
"... Abstract We consider CSP from the point of view of the algebraic theory of effects, which classifies operations as effect constructors and effect deconstructors; it also provides a link with functional programming, being a refinement of Moggi’s seminal monadic point of view. There is a natural algeb ..."
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Abstract We consider CSP from the point of view of the algebraic theory of effects, which classifies operations as effect constructors and effect deconstructors; it also provides a link with functional programming, being a refinement of Moggi’s seminal monadic point of view. There is a natural algebraic theory of the constructors whose free algebra functor is Moggi’s monad; we illustrate this by characterising free and initial algebras in terms of two versions of the stable failures model of CSP, one more general than the other. Deconstructors are dealt with as homomorphisms to (possibly nonfree) algebras. One can view CSP’s action and nondeterminism operators as constructors and the rest, such as concealment and concurrency, as deconstructors. Carrying this programme out results in taking deterministic external choice as constructor rather than general external choice. However, binary deconstructors, such as the CSP concurrency operator, provide unresolved difficulties. We conclude by presenting a combination of CSP with Moggi’s computational λcalculus, in which the operators, including concurrency, are polymorphic. While the paper mainly concerns CSP, it ought to be possible to carry over similar ideas to other process calculi. Rob van Glabbeek
Semidecidability of may, must . . . in a highertype setting
, 2009
"... We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic pro ..."
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We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semidecidability of similar tests under the simultaneous presence of nondeterministic and probabilistic choice. Keywords: Nondeterministic and probabilistic computation, highertype computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains.
OBSERVATIONALLYINDUCED ALGEBRAS IN DOMAIN THEORY
, 2014
"... Vol. 10(3:18)2014, pp. 1–26 www.lmcsonline.org ..."
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