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Comparing hierarchies of total functionals
 Logical Methods in Computer Science, Volume 1, Issue 2, Paper 4 (2005). RICH HIERARCHY 21
"... In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem i ..."
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In this paper, we will address a problem raised by Bauer, Escardó and Simpson. We define two hierarchies of total, continuous functionals over the reals based on domain theory, one based on an “extensional ” representation of the reals and the other on an “intensional ” representation. The problem is if these two hierarchies coincide. We will show that this coincidence problem is equivalent to the statement that the topology on the KleeneKreisel continuous functionals of a fixed type induced by all continuous functions into the reals is zerodimensional for each type. As a tool of independent interest, we will construct topological embeddings of the KleeneKreisel functionals into both the extensional and the intensional hierarchy at each type. The embeddings will be hierarchy embeddings as well in the sense that they are the inclusion maps at type 0 and respect application at higher types. 1
Realizability Models Refuting Ishihara’s Boundedness Principle
, 2011
"... In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN ..."
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In [Ish92] H. Ishihara introduced the socalled boundedness principle BDN which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BDN is
mlq header will be provided by the publisher On Local NonCompactness in Recursive Mathematics
"... theory ..."
Realizability Models Refuting Ishihara’s Boundedness Principle
"... Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We constr ..."
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Ishihara’s Boundedness Principle BDN was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from N N to N is sequentially continuous and in which the axiom of choice from N N to N holds. Since the latter is known to be inconsistent with the statement that all functions from N N to N are continuous these models refute BDN.
Final Report for EPSRC Grant GR/S46710/01 Topological Models for Computational Metalanguages Background and context
"... Denotational semantics seeks to provide compositional models of computation, programs and data at the right level of abstraction to be independent of languagespecific and implementation details, and domain theory provides the mathematical objects needed to construct such models. Although domain the ..."
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Denotational semantics seeks to provide compositional models of computation, programs and data at the right level of abstraction to be independent of languagespecific and implementation details, and domain theory provides the mathematical objects needed to construct such models. Although domain theory is a rich and highly developed mathematical theory, in its traditional form it fails to provide models for a number of significant computational phenomena, especially in combination. For example, although domain theory is able to model any two of: function types (as used in functional languages), probabilistic choice (as used in randomised algorithms) and computability (i.e., the requirement that programs are actually given by algorithms), it is not known how to model all three in combination. This research project undertook the study of a generalised domain theory based on widening the mathematical interpretation of the notion of “domain ” to include a broader collection of topological spaces than usually considered. (Traditional “domains ” can be viewed as topological spaces under their associated “Scott topology”.) The principal goal was to establish the main mathematical properties of the associated “topological domain theory”, and to demonstrate its ability to overcome limitations of traditional domain theory. That such a programme should be possible was suggested by the PI’s observation that “topological domains” arise naturally as a model for an abstract logicbased approach to domain theory known as “synthetic domain