Turing's unpublished algorithm for normal numbers

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In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he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 e...

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Detalles Bibliográficos
Autores principales: Becher, V., Figueira, S., Picchi, R.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2007
Materias:
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Turing's unpublished manuscript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
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paperaa:paper_03043975_v377_n1-3_p126_Becher
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spelling paperaa:paper_03043975_v377_n1-3_p126_Becher2023-06-12T16:47:27Z Turing's unpublished algorithm for normal numbers Theor Comput Sci 2007;377(1-3):126-138 Becher, V. Figueira, S. Picchi, R. Algorithm for normal numbers Computable absolutely normal numbers Turing's unpublished manuscript Error correction Number theory Set theory Theorem proving Turing machines Computable construction Lebesgue measure Manuscripts Normal numbers Algorithms In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he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. 2007 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
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
spellingShingle 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
Becher, V.
Figueira, S.
Picchi, R.
Turing's unpublished algorithm for normal numbers
topic_facet 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
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.
format Artículo
Artículo
publishedVersion
author Becher, V.
Figueira, S.
Picchi, R.
author_facet Becher, V.
Figueira, S.
Picchi, R.
author_sort Becher, V.
title Turing's unpublished algorithm for normal numbers
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
publishDate 2007
url http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher
work_keys_str_mv AT becherv turingsunpublishedalgorithmfornormalnumbers
AT figueiras turingsunpublishedalgorithmfornormalnumbers
AT picchir turingsunpublishedalgorithmfornormalnumbers
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