Turing's unpublished algorithm for normal numbers
In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he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...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | Argentina |
| Institución: | Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales |
| Repositorio: | Biblioteca Digital (UBA-FCEN) |
| Idioma: | inglés |
| OAI Identifier: | paperaa:paper_03043975_v377_n1-3_p126_Becher |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_03043975_v377_n1-3_p126_Becher |
| Access Level: | acceso abierto |
| Palabra clave: | Algorithm for normal numbers Computable absolutely normal numbers Turing's unpublished manuscript Error correction Number theory Set theory Theorem proving Turing machines Computable construction Lebesgue measure Manuscripts Normal numbers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 |
| Sumario: | In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he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. |
|---|