450 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
A rich hierarchy of functionals of finite types
We are considering typed hierarchies of total, continuous functionals using
complete, separable metric spaces at the base types. We pay special attention
to the so called Urysohn space constructed by P. Urysohn. One of the properties
of the Urysohn space is that every other separable metric space can be
isometrically embedded into it. We discuss why the Urysohn space may be
considered as the universal model of possibly infinitary outputs of algorithms.
The main result is that all our typed hierarchies may be topologically
embedded, type by type, into the corresponding hierarchy over the Urysohn
space. As a preparation for this, we prove an effective density theorem that is
also of independent interest.Comment: 21 page
Inductive Definition and Domain Theoretic Properties of Fully Abstract
A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF +
"parallel conditional function"), respectively, is presented. It is based on
general notions of sequential computational strategies and wittingly consistent
non-deterministic strategies introduced by the author in the seventies.
Although these notions of strategies are old, the definition of the fully
abstract models is new, in that it is given level-by-level in the finite type
hierarchy. To prove full abstraction and non-dcpo domain theoretic properties
of these models, a theory of computational strategies is developed. This is
also an alternative and, in a sense, an analogue to the later game strategy
semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong;
and Nickau. In both cases of PCF and PCF^+ there are definable universal
(surjective) functionals from numerical functions to any given type,
respectively, which also makes each of these models unique up to isomorphism.
Although such models are non-omega-complete and therefore not continuous in the
traditional terminology, they are also proved to be sequentially complete (a
weakened form of omega-completeness), "naturally" continuous (with respect to
existing directed "pointwise", or "natural" lubs) and also "naturally"
omega-algebraic and "naturally" bounded complete -- appropriate generalisation
of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page
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