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18
The troublesome probabilistic powerdomain
 Proceedings of the Third Workshop on Computation and Approximation
, 1998
"... In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerd ..."
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Cited by 42 (5 self)
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In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerdomain construction can be restricted to RB. Inthispaper, wegiveacounterexampletoGraham’sproofanddescribe our own attempts at proving a closure result for the probabilistic powerdomain construction. We have positive results for finite trees and finite reversed trees. These illustrate the difficulties we face, rather than being a satisfying answer to the question of whether the probabilistic powerdomain and function spaces can be reconciled. We are more successful with coherent or Lawsoncompact domains. These form a category with many pleasing properties but they fall short of supporting function spaces. Along the way, we give a new proof of Jones ’ Splitting Lemma. 1
Probabilistic Domains
 in Proc. CAAP ’94, LNCS
, 1997
"... We show the equivalence of several different axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a finite number of possibilities with arbitr ..."
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Cited by 22 (4 self)
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We show the equivalence of several different axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a finite number of possibilities with arbitrary probabilities, whereas the poorest one has binary choice with equal probabilities as the only operation. The remaining theories lie in between; one of them is the theory of binary choice by Graham [1]. 1 Introduction A probabilistic programming language could contain different kinds of language constructs to express probabilistic choice. In a rather poor language, there might be a construct x \Phi y, whose semantics is a choice between the two possibilities x and y with equal probabilities 1=2. The `possibilities' x and y can be statements in an imperative language or expressions in a functional language. A quite rich language could contain a construct [p 1 : x 1 ; : : : ; p n : x n ],...
The regularlocallycompact coreflection of stably locally compact locale
 Journal of Pure and Applied Algebra
, 2001
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally comp ..."
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Cited by 18 (9 self)
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The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
"... ..."
On the duality of compact vs. open
 Papers on General Topology and Applications: Eleventh Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences
, 1996
"... It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We presen ..."
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Cited by 13 (1 self)
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It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We present a similar approach to describe the Stoneduals of coherent spaces, thus dropping the requirement of having a base of compactopens (or, alternatively, replacing algebraicity of the lattices by continuity). The construction via strong proximity lattices is resembling the classical case, just replacing the order by an order of approximation. Our development enlightens the fact that “open ” and “compact ” are dual concepts which merely happen to coincide in the classical case.
Topologies on spaces of continuous functions
 Topology Proc
"... It is wellknown that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compactopen topology. It is less wellknown that among arbitrary topological spaces, the exponentiable spaces are precisely the corecompact spaces. T ..."
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Cited by 5 (1 self)
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It is wellknown that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compactopen topology. It is less wellknown that among arbitrary topological spaces, the exponentiable spaces are precisely the corecompact spaces. The available approaches to the general characterization are based on either category theory or continuouslattice theory, or even both. It is the main purpose of this paper to provide a selfcontained, elementary and brief development of general function spaces. The only prerequisite to this development is a basic knowledge of general topology (continuous functions, product topology and compactness). But another connection with the theory of continuous lattices lurks in this approach to function spaces, which is examined after the elementary exposition is completed. Continuity of the functionevaluation map is shown to coincide with a certain approximation property of a topology on the frame of open sets of the exponent space, and the existence of a smallest approximating topology is equivalent to exponentiability of the space. We show that the intersection of the approximating topologies of any preframe is the Scott topology. In particular, we conclude that a complete lattice is continuous if and only if it has a smallest approximating topology and finite meets distribute over directed joins. 1
A DomainTheoretic BanachAlaoglu Theorem, Festschrift for Klaus Keimel
 MSCS
"... Abstract. We give a domaintheoretic analogue of the classical BanachAlaoglu theorem, showing that the patch topology on the weak * topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the p ..."
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Cited by 5 (1 self)
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Abstract. We give a domaintheoretic analogue of the classical BanachAlaoglu theorem, showing that the patch topology on the weak * topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set ’ of linear functionals is compact. 1
An Upper Power Domain Construction in terms of Strongly Compact Sets
 MFPS '91. LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... A novel upper power domain construction is defined by means of strongly compact sets. Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk. The notion of strong compactness allows a proof of st ..."
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Cited by 4 (3 self)
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A novel upper power domain construction is defined by means of strongly compact sets. Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk. The notion of strong compactness allows a proof of stronger properties than compactness would, e.g. an intrinsic universal property of the upper power construction, and its commutation with the lower construction.
On the CompactRegular Coreflection of a Stably Compact Locale
"... A nucleus on a frame is a finitemeet preserving closure operator. The nuclei on a frame form themselves a frame, with the Scott continuous nuclei as a subframe. We refer to this subframe as the patch frame. We show that the patch construction exhibits the category of compact regular locales and con ..."
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Cited by 3 (2 self)
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A nucleus on a frame is a finitemeet preserving closure operator. The nuclei on a frame form themselves a frame, with the Scott continuous nuclei as a subframe. We refer to this subframe as the patch frame. We show that the patch construction exhibits the category of compact regular locales and continuous maps as a coreflective subcategory of the category of stably compact locales and perfect maps, and the category of Stone locales and continuous maps as a coreflective subcategory of the category of spectral locales and spectral maps. We relate our patch construction to Banaschewski and Brummer's construction of the dual equivalence of the category of stably compact locales and perfect maps with the category of compact regular biframes and biframe homomorphisms. Keywords: Frame of nuclei, Scott continuous nuclei, patch topology, stably locally compact locales, perfect maps, compact regular locales. AMS Classification: 06A15, 06B35, 06D20, 06E15, 54C10, 54D45, 54F05. 1 Introduction ...
Power Domains Supporting Recursion and Failure
 In CAAP'92
, 1998
"... Following the program of Moggi, the semantics of a simple nondeterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle nontermination properly. Instead, a novel con ..."
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Cited by 3 (1 self)
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Following the program of Moggi, the semantics of a simple nondeterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle nontermination properly. Instead, a novel construction is proposed and investigated. It embodies both nondeterminism (choice and failure) and possible nontermination caused by recursion. 1 Introduction Following the proposals of Moggi [Mog89, Mog91b], functional languages with various notions of computations can be denotationally described by means of monads. Monads are constructions mapping domains of values into domains of computations for these values. Computations involving destructive assignments, for instance, are handled by the state transformer monad [Wad90], whereas computations by nondeterministic choice are handled by power domain constructions [Plo76, Smy78, Gun90]. All known power domain constructions and many others ar...