A topological and domain theoretical study of total computable functions
Topologically the set of total computable functions has been studied only as a subspace of a Baire space. Where the topology of this Baire space is the induced topology of a Scott topology for the partial functions (not necessarily computable). In this thesis a novel topology on the index set of...
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| Formato: | doctoralThesis |
| Idioma: | por |
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| Endereço do item: | https://repositorio.ufrn.br/jspui/handle/123456789/22327 |
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| Resumo: | Topologically the set of total computable functions has been studied only as
a subspace of a Baire space. Where the topology of this Baire space is the
induced topology of a Scott topology for the partial functions (not necessarily
computable). In this thesis a novel topology on the index set of the set of total
computable functions is built and proved that it is not homeomorphic to the
aforementioned subspace of the presented Baire space. There is an important
undecidable subset of the set of total computable functions called the set of
regular computable functions that receives particular attention in this thesis.
In order to make a topological study of this set a whole theoretical apparatus
is constructed. After presenting the state of the art of generalised domain
theory, a novel generalisation of algebraic domains coined as algebraic quasidomains
is introduced. With a suited partial-order an algebraic quasi-domain
is built for the set of total computable functions. Through the Scott topology
associated with this algebraic quasi-domain a necessary condition for the regular
computable functions is obtained. It is proved that this later topology is not
homeomorphic to the previously mentioned subspace of the presented Baire
space. As a byproduct a Scott topology for the set of total functions (not
necessarily computable) is introduced. It is proved that this later topology is
not homeomorphic to the presented Baire space. It is also proved that the Scott topologies in the set of total functions and in the subset of total computable functions have the set of total functions with
finite support as a common dense set. Analogously it is proved that the topology
in the index set of the set of total computable functions has as a dense set the
indexes corresponding to a computable enumeration without repetition of the
set of total functions with finite support. |
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