Results 1  10
of
36
Domain Representations of Topological Spaces
, 2000
"... A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, for example, discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors. It is shown that any T 0 topological space has a domain representation. These domain representations are very large. However, smaller domain representations are also constructed for large classes of spaces. For example, each second countable regular Hausdorff space has a domain representation with a countable base. Domain representations of functions and function spaces are also studied.
Lazy Functional Algorithms for Exact Real Functionals
 Lec. Not. Comput. Sci
, 1998
"... . We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a ..."
Abstract

Cited by 32 (0 self)
 Add to MetaCart
(Show Context)
. We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory. 1 Introduction In exact real number computation, infinite representations of reals are employed to avoid the usual rounding errors that are inherent in floating point computation [46, 17]. For certain real number computations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floatingpoint algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation
, 1996
"... ..."
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of thi ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
(Show Context)
A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
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 ..."
Abstract

Cited by 21 (8 self)
 Add to MetaCart
(Show Context)
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,
Injective spaces via the filter monad
 Topology Proceedings
, 1997
"... An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein typ ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
(Show Context)
An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein type and apply this to obtain a simple proof of the fact that the algebras are the continuous lattices (Alan Day, 1975, Oswald Wyler, 1976). In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras. For the filter monad, the class turns out to consist precisely of the subspace embeddings. We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972). Similar results are obtained for continuous Scott domains, which are characterized as the injective spaces over dense subspace embeddings. Keywords: Extension of maps, injective space, continuous lattice, continuous Scott domain, domain theory, KockZöberlein monad. AMS classification: 54C20, 06B35, 18C20. 1
Polish spaces, computable approximations, and bitopological spaces
, 2000
"... Answering a question of J. Lawson (formulated also earlier, in 1984, by Kamimura and Tang [16]) we show that every Polish space admits a bounded complete computational model, as defined below. This results from our construction, in each Polish space 〈X, τ〉, of a countable family C of nonempty close ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Answering a question of J. Lawson (formulated also earlier, in 1984, by Kamimura and Tang [16]) we show that every Polish space admits a bounded complete computational model, as defined below. This results from our construction, in each Polish space 〈X, τ〉, of a countable family C of nonempty closed subsets of X such that: (cp) each subset of C with the finite intersection property has nonempty intersection;
Induction and recursion on the partial real line with applications to Real PCF
 Theoretical Computer Science
, 1997
"... The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the socalled Peano axioms for natural numbers. The theory is based on a domainequationlike presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higherorder functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...
On the compactregular coreflection of a stably compact locale
 MFPS XV PRELIMINARY VERSION
"... 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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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