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Previous issue date: 2016-07-29 / O conjunto de fun??es totais comput?veis somente tem sido estudado topologicamente como um subespa?o de um espa?o de Baire. Onde a topologia deste espa?o de Baire ? a topologia induzida de uma topologia de Scott para as fun??es parciais (n?o necessariamente comput?veis). Nesta tese constr?i-se uma topologia original no conjunto de ?ndices das fun??es totais comput?veis e demonstra-se que ela n?o ? homeomorfa com o subespa?o do espa?o de Baire que foi mencionado. H? um subconjunto indecid?vel importante no conjunto de fun??es totais comput?veis chamado ?o conjunto de fun??es comput?veis regulares?, que recebe aten??o especial nesta tese. Com a finalidade de fazer um estudo topol?gico deste conjunto constr?i-se todo um aparato te?rico. Ap?s apresentar o estado da arte da teoria dos dom?nios generalizada introduz-se uma generaliza??o original dos dom?nios alg?bricos nomeados como ?quase dom?nios alg?bricos?. Com uma ordem parcial adequada, constr?i-se um quase-dom?nio alg?brico para o conjunto de fun??es comput?veis totais. Por meio da topologia de Scott associada a esse quase-dom?nio alg?brico, obt?m-se uma condi??o necess?ria para as fun??es comput?veis regulares. Fica provado que esta ?ltima topologia n?o ? homeomorfa com o subespa?o previamente mencionado do espa?o de Baire apresentado. Como subproduto, introduz-se uma topologia de Scott para o conjunto de fun??es totais (n?o necessariamente comput?veis). Fica provado que esta ?ltima topologia n?o ? homeomorfa com o espa?o de Baire apresentado. Fica tamb?m provado que as topologias de Scott no conjunto de fun??es totais e no subconjunto de fun??es totais comput?veis t?m o conjunto de fun??es totais com suporte finito como conjunto denso comum. Analogamente, fica provado que a topologia no conjunto ?ndice do conjunto de fun??es totais comput?veis tem como conjunto denso os ?ndices correspondentes a uma enumera??o comput?vel sem repeti??o do conjunto de fun??es totais com suporte infinito. / 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.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.ufrn.br:123456789/22327 |
| Date | 29 July 2016 |
| Creators | Olguin, Cl?udio Andr?s Callejas |
| Contributors | 90688384404, http://lattes.cnpq.br/4601263005352005, Santiago, Regivan Hugo Nunes, 30680581200, http://lattes.cnpq.br/7536988783793885, Santana, Fagner Lemos de, 00968711480, http://lattes.cnpq.br/9444112594388983, Viana, Jorge Petrucio, 84409967720, http://lattes.cnpq.br/2332327416299928, Oliveira J?nior, Wilson Rosa de, 27731715434, http://lattes.cnpq.br/1812805997541734, Bedregal, Benjamin Rene Callejas |
| Publisher | PROGRAMA DE P?S-GRADUA??O EM SISTEMAS E COMPUTA??O, UFRN, Brasil |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/doctoralThesis |
| Source | reponame:Repositório Institucional da UFRN, instname:Universidade Federal do Rio Grande do Norte, instacron:UFRN |
| Rights | info:eu-repo/semantics/openAccess |
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