Fractal Dimension versus Process Complexity

We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine and any particular input , we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of...

Descripción completa

Detalles Bibliográficos
Autores: Joosten, Joost J., Soler-Toscano, Fernando, Zenil, Hector
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2016
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/106693
Acceso en línea:https://hdl.handle.net/2445/106693
Access Level:acceso abierto
Palabra clave:Lògica matemàtica
Filosofia de la matemàtica
Fractals
Mathematical logic
Philosophy of mathematics
Turing, Alan Mathison, 1912-1954
id ES_708d38bf92dcb7d35b7cd2e25e19fea8
oai_identifier_str oai:diposit.ub.edu:2445/106693
network_acronym_str ES
network_name_str España
repository_id_str
spelling Fractal Dimension versus Process ComplexityJoosten, Joost J.Soler-Toscano, FernandoZenil, HectorLògica matemàticaFilosofia de la matemàticaFractalsMathematical logicPhilosophy of mathematicsFractalsTuring, Alan Mathison, 1912-1954We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine and any particular input , we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation . In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in time , we have empirically verified that the corresponding dimension is , a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.Hindawi2016info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/106693Articles publicats en revistes (Filosofia)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: https://doi.org/10.1155/2016/5030593Advances in Mathematical Physics, 2016, p. 1-21https://doi.org/10.1155/2016/5030593cc-by (c) Joosten, Joost J. et al., 2016http://creativecommons.org/licenses/by/3.0/esinfo:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1066932026-05-27T06:46:51Z
dc.title.none.fl_str_mv Fractal Dimension versus Process Complexity
title Fractal Dimension versus Process Complexity
spellingShingle Fractal Dimension versus Process Complexity
Joosten, Joost J.
Lògica matemàtica
Filosofia de la matemàtica
Fractals
Mathematical logic
Philosophy of mathematics
Fractals
Turing, Alan Mathison, 1912-1954
title_short Fractal Dimension versus Process Complexity
title_full Fractal Dimension versus Process Complexity
title_fullStr Fractal Dimension versus Process Complexity
title_full_unstemmed Fractal Dimension versus Process Complexity
title_sort Fractal Dimension versus Process Complexity
dc.creator.none.fl_str_mv Joosten, Joost J.
Soler-Toscano, Fernando
Zenil, Hector
author Joosten, Joost J.
author_facet Joosten, Joost J.
Soler-Toscano, Fernando
Zenil, Hector
author_role author
author2 Soler-Toscano, Fernando
Zenil, Hector
author2_role author
author
dc.subject.none.fl_str_mv Lògica matemàtica
Filosofia de la matemàtica
Fractals
Mathematical logic
Philosophy of mathematics
Fractals
Turing, Alan Mathison, 1912-1954
topic Lògica matemàtica
Filosofia de la matemàtica
Fractals
Mathematical logic
Philosophy of mathematics
Fractals
Turing, Alan Mathison, 1912-1954
description We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine and any particular input , we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation . In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in time , we have empirically verified that the corresponding dimension is , a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.
publishDate 2016
dc.date.none.fl_str_mv 2016
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/106693
url https://hdl.handle.net/2445/106693
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1155/2016/5030593
Advances in Mathematical Physics, 2016, p. 1-21
https://doi.org/10.1155/2016/5030593
dc.rights.none.fl_str_mv cc-by (c) Joosten, Joost J. et al., 2016
http://creativecommons.org/licenses/by/3.0/es
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by (c) Joosten, Joost J. et al., 2016
http://creativecommons.org/licenses/by/3.0/es
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Hindawi
publisher.none.fl_str_mv Hindawi
dc.source.none.fl_str_mv Articles publicats en revistes (Filosofia)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869410588995616768
score 15,300724

Similares en LA Referencia