Results 1  10
of
90
Measure, Randomness and Sublocales
"... This paper investigates aspects of measure and randomness in the context of locale theory (pointfree topology). We prove that every measure (σcontinuous valuation) µ, on the σframe of opens of a fitted σlocale X, extends to a measure on the lattice of all σsublocales of X (Theorem 1). Furthermo ..."
Abstract
 Add to MetaCart
This paper investigates aspects of measure and randomness in the context of locale theory (pointfree topology). We prove that every measure (σcontinuous valuation) µ, on the σframe of opens of a fitted σlocale X, extends to a measure on the lattice of all σsublocales of X (Theorem 1
Properly injective spaces and function spaces
 TO APPEAR IN TOPOLOGY AND ITS APPLICATIONS
, 1997
"... Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous ..."
Abstract

Cited by 36 (12 self)
 Add to MetaCart
Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial onepoint space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meetsemilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in posetenriched categories with the structure of a KZmonad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.
Étale groupoids and their quantales
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
Abstract

Cited by 34 (9 self)
 Add to MetaCart
We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
Abstract

Cited by 32 (4 self)
 Add to MetaCart
). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map
Localic completion of generalized metric spaces II: Powerlocales
, 2009
"... The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial resu ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann. As applications, a localic completion is always overt, and is compact iff its generalized metric space is totally bounded. The representation is used to discuss closed intervals of the reals, with the localic Heine–Borel Theorem as a consequence. The work is constructive in the toposvalid sense.
The Ektara architecture: The right framework for contextaware wearable and ubiquitous computing applications
, 2000
"... In this paper we describe the Ektara architecture, a distributed computing architecture for building contextaware ubiquitous and wearable computing applications (UWC). We begin by describing the critical requirements for developing real contextaware UWC applications and relate these to a plausible ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
In this paper we describe the Ektara architecture, a distributed computing architecture for building contextaware ubiquitous and wearable computing applications (UWC). We begin by describing the critical requirements for developing real contextaware UWC applications and relate these to a plausible usercentered scenario. We then present the functional components of the Ektara architecture and explain how they address the critical requirements. Examples of how these functional components interact to create real applications are given, and we discuss our progress in implementing a prototype system and several applications.
Information Systems for Continuous Posets
, 1993
"... The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous S ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's "Rstructures". Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe HoffmannLawson duality (which could not be done easily with Rstructures) and Vietoris power locales. 2 We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets, Vietoris algebras are the same as localic semilattices.
Interweaving Reason, Action and Perception
, 1991
"... In an attempt to understand and emulate intelligent behavior Artificial Intelligence researchers have historically taken a reductionist approach and divided their investigation into separate studies of reason, perception and action. As a consequence, intelligent robots have been constructed using a ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
In an attempt to understand and emulate intelligent behavior Artificial Intelligence researchers have historically taken a reductionist approach and divided their investigation into separate studies of reason, perception and action. As a consequence, intelligent robots have been constructed using a coarse grained architecture; reasoning, perception and action have been implemented as separate modules that interact infrequently. This paper describes an investigation into the effect of reducing this architecture granularity. It shows that significant computational efficiencies can be gained by introducing a fine grainedintegration or "interweaving" of these functions and demonstrates these savings for an intelligent navigation system. The paper introduces the "reason a little, move a little, look a little" paradigm (RML), describes an RML implementation, presents analytical arguments that show the RML scheme can result in significant reduction in complexity for both planning and vision,...
Computably based locally compact spaces
, 2003
"... ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalen ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth’s effectively given domains and Jung’s Strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the waybelow relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott’s domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.
Probabilistic Power Domains, Information Systems, and Locales
 Mathematical Foundations of Programming Semantics VIII, volume 802 of Lecture Notes in Computer Science
, 1994
"... The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems `a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide. 1 Introduction ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems `a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide. 1 Introduction
Results 1  10
of
90