Turing's unpublished algorithm for normal numbers

Autores
Becher, V.; Figueira, S.; Picchi, R.
Año de publicación
2007
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result. © 2007 Elsevier Ltd. All rights reserved.
Fil:Becher, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Figueira, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fuente
Theor Comput Sci 2007;377(1-3):126-138
Materia
Algorithm for normal numbers
Computable absolutely normal numbers
Turing's unpublished manuscript
Error correction
Number theory
Set theory
Theorem proving
Turing machines
Computable construction
Lebesgue measure
Manuscripts
Normal numbers
Algorithms
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by/2.5/ar
Repositorio
Biblioteca Digital (UBA-FCEN)
Institución
Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
OAI Identificador
paperaa:paper_03043975_v377_n1-3_p126_Becher
Acceder
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oai_identifier_str paperaa:paper_03043975_v377_n1-3_p126_Becher
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network_name_str Biblioteca Digital (UBA-FCEN)
spelling Turing's unpublished algorithm for normal numbersBecher, V.Figueira, S.Picchi, R.Algorithm for normal numbersComputable absolutely normal numbersTuring's unpublished manuscriptError correctionNumber theorySet theoryTheorem provingTuring machinesComputable constructionLebesgue measureManuscriptsNormal numbersAlgorithmsIn an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result. © 2007 Elsevier Ltd. All rights reserved.Fil:Becher, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Figueira, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.2007info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_BecherTheor Comput Sci 2007;377(1-3):126-138reponame:Biblioteca Digital (UBA-FCEN)instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesinstacron:UBA-FCENenginfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/ar2025-09-04T09:48:35Zpaperaa:paper_03043975_v377_n1-3_p126_BecherInstitucionalhttps://digital.bl.fcen.uba.ar/Universidad públicaNo correspondehttps://digital.bl.fcen.uba.ar/cgi-bin/oaiserver.cgiana@bl.fcen.uba.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:18962025-09-04 09:48:36.608Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv Turing's unpublished algorithm for normal numbers
title Turing's unpublished algorithm for normal numbers
spellingShingle Turing's unpublished algorithm for normal numbers
Becher, V.
Algorithm for normal numbers
Computable absolutely normal numbers
Turing's unpublished manuscript
Error correction
Number theory
Set theory
Theorem proving
Turing machines
Computable construction
Lebesgue measure
Manuscripts
Normal numbers
Algorithms
title_short Turing's unpublished algorithm for normal numbers
title_full Turing's unpublished algorithm for normal numbers
title_fullStr Turing's unpublished algorithm for normal numbers
title_full_unstemmed Turing's unpublished algorithm for normal numbers
title_sort Turing's unpublished algorithm for normal numbers
dc.creator.none.fl_str_mv Becher, V.
Figueira, S.
Picchi, R.
author Becher, V.
author_facet Becher, V.
Figueira, S.
Picchi, R.
author_role author
author2 Figueira, S.
Picchi, R.
author2_role author
author
dc.subject.none.fl_str_mv Algorithm for normal numbers
Computable absolutely normal numbers
Turing's unpublished manuscript
Error correction
Number theory
Set theory
Theorem proving
Turing machines
Computable construction
Lebesgue measure
Manuscripts
Normal numbers
Algorithms
topic Algorithm for normal numbers
Computable absolutely normal numbers
Turing's unpublished manuscript
Error correction
Number theory
Set theory
Theorem proving
Turing machines
Computable construction
Lebesgue measure
Manuscripts
Normal numbers
Algorithms
dc.description.none.fl_txt_mv In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result. © 2007 Elsevier Ltd. All rights reserved.
Fil:Becher, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Figueira, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
description In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing's ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turing's proof idea and obtain his result. © 2007 Elsevier Ltd. All rights reserved.
publishDate 2007
dc.date.none.fl_str_mv 2007
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher
url http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/2.5/ar
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/2.5/ar
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv Theor Comput Sci 2007;377(1-3):126-138
reponame:Biblioteca Digital (UBA-FCEN)
instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron:UBA-FCEN
reponame_str Biblioteca Digital (UBA-FCEN)
collection Biblioteca Digital (UBA-FCEN)
instname_str Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron_str UBA-FCEN
institution UBA-FCEN
repository.name.fl_str_mv Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
repository.mail.fl_str_mv ana@bl.fcen.uba.ar
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score 12.623145

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