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12
A Semantic Model for Graphical User Interfaces
, 2011
"... We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capt ..."
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Cited by 8 (1 self)
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We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capture the arbitrariness of user input (e.g., a user gets to decide the stream of clicks she sends to a program) by making use of the fact that the closed subsets of a metric space themselves form a metric space under the Hausdorff metric, allowing us to interpret nondeterminism with a “powerspace ” monad on ultrametric spaces. The powerspace monad is commutative, and hence gives rise to a model of linear logic. We exploit this fact by constructing a mixed linear/nonlinear domainspecific language for GUI programming. The linear sublanguage naturally captures the usage constraints on the various linear objects in GUIs, such as the elements of a DOM or scene graph. We have implemented this DSL as an extension to OCaml, and give examples demonstrating that programs in this style can be short and readable.
A constructive and functorial embedding of locally compact metric spaces
, 2006
"... The paper establishes, within constructive mathematics, a full and faithful ..."
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Cited by 7 (3 self)
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The paper establishes, within constructive mathematics, a full and faithful
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Cited by 4 (0 self)
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
Resolution of the uniform lower bound problem in constructive analysis
, 2007
"... constructive analysis ..."
COMPUTABLE SETS: LOCATED AND OVERT LOCALES
, 2007
"... Abstract. What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions intr ..."
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Abstract. What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in locale theory in a constructive, or topos theoretic, context. We show that the two notions are intimately connected. We propose a definition of located closed sublocale motivated by locatedness of subsets of metric spaces. A closed sublocale of a compact regular locale is located iff it is overt. Moreover, a closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. For Baire space metric locatedness corresponds to having a decidable positivity predicate. Finally, we show that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all are results are valid in any predicative topos. 1.
LOCATEDNESS AND OVERT SUBLOCALES
, 2009
"... Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected. Bishop defines a ..."
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Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected. Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact ’ is translated as compact and overt. We propose a definition of located predicate on subspaces in formal topology. We call a sublocale located if it can be presented by a formal topology with a located predicate. We prove that a closed sublocale of a compact regular locale has a located predicate iff it is overt. Moreover, a Bishopclosed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all our results are valid in any predicative topos.
Gelfand spectra in Grothendieck toposes using geometric mathematics
, 1310
"... In the topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to any unital C*algebra A, a topos T (A) of sheaves on a locale and a commutative C*algebra A within that topos. The Gelfand spectrum of A is a locale Σ in this topos, which is equivalent to a bundle over the ..."
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In the topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to any unital C*algebra A, a topos T (A) of sheaves on a locale and a commutative C*algebra A within that topos. The Gelfand spectrum of A is a locale Σ in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale Σ, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise. As a byproduct of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too. 1
U.U.D.M. Report 2009:1 Open sublocales of localic completions
, 2009
"... We give a constructive characterization of morphisms between open sublocales of localic completions of locally compact metric (LCM) spaces, in terms of continuous functions. The category of open subspaces of LCM spaces is thereby shown to embed fully faithfully into ..."
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We give a constructive characterization of morphisms between open sublocales of localic completions of locally compact metric (LCM) spaces, in terms of continuous functions. The category of open subspaces of LCM spaces is thereby shown to embed fully faithfully into
Applications of geometric logic to topos
, 2009
"... This document describes my 3year project “Applications of geometric logic to topos approaches to quantum theory”, to start in 2009 with funding from the UK Engineering and Physical Sciences Research Council (EPSRC) for a postdoctoral Research Assistant and a PhD studentship. After an overview of t ..."
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This document describes my 3year project “Applications of geometric logic to topos approaches to quantum theory”, to start in 2009 with funding from the UK Engineering and Physical Sciences Research Council (EPSRC) for a postdoctoral Research Assistant and a PhD studentship. After an overview of the background and programme of work, it leads on to a description of the highgrade postdoctoral post funded as part of the project. 1