On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a...

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Autores: Estevan Muguerza, Asier, Miñana, Juan José, Valero, Óscar
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2019
País:España
Institución:Universidad Pública de Navarra
Repositorio:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
OAI Identifier:oai:academica-e.unavarra.es:2454/36069
Acceso en línea:https://hdl.handle.net/2454/36069
Access Level:acceso abierto
Palabra clave:Partial order
Quasi-metric
Fixed point
Kleene
Asymptotic complexity
Recurrence equation
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spelling On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithmsEstevan Muguerza, AsierMiñana, Juan JoséValero, ÓscarPartial orderQuasi-metricFixed pointKleeneAsymptotic complexityRecurrence equationThe celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.A. Estevan acknowledges financial support from Spanish Ministry of Economy and Competitiveness under Grants MTM2015-63608-P (MINECO/FEDER) and ECO2015-65031. J.J. Miñana and O. Valero acknowledge financial support from Spanish Ministry of Science, Innovation and Universities under Grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds. This work is also partially supported by Programa Operatiu FEDER 2014–2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d’Innovació i Recerca, Govern de les Illes Balears) and by project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776.SpringerEstatistika, Informatika eta MatematikaInstitute for Advanced Materials and Mathematics - INAMAT2Estadística, Informática y Matemáticas2019info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2454/36069reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarrainstname:Universidad Pública de NavarraInglésinfo:eu-repo/grantAgreement/MINECO//MTM2015-63608-Pinfo:eu-repo/grantAgreement/European Commission/Horizon 2020 Framework Programme/779776© The Royal Academy of Sciences, Madrid 2019info:eu-repo/semantics/openAccessoai:academica-e.unavarra.es:2454/360692026-06-17T12:41:47Z
dc.title.none.fl_str_mv On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
title On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
spellingShingle On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
Estevan Muguerza, Asier
Partial order
Quasi-metric
Fixed point
Kleene
Asymptotic complexity
Recurrence equation
title_short On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
title_full On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
title_fullStr On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
title_full_unstemmed On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
title_sort On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms
dc.creator.none.fl_str_mv Estevan Muguerza, Asier
Miñana, Juan José
Valero, Óscar
author Estevan Muguerza, Asier
author_facet Estevan Muguerza, Asier
Miñana, Juan José
Valero, Óscar
author_role author
author2 Miñana, Juan José
Valero, Óscar
author2_role author
author
dc.contributor.none.fl_str_mv Estatistika, Informatika eta Matematika
Institute for Advanced Materials and Mathematics - INAMAT2
Estadística, Informática y Matemáticas
dc.subject.none.fl_str_mv Partial order
Quasi-metric
Fixed point
Kleene
Asymptotic complexity
Recurrence equation
topic Partial order
Quasi-metric
Fixed point
Kleene
Asymptotic complexity
Recurrence equation
description The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.
publishDate 2019
dc.date.none.fl_str_mv 2019
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2454/36069
url https://hdl.handle.net/2454/36069
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MINECO//MTM2015-63608-P
info:eu-repo/grantAgreement/European Commission/Horizon 2020 Framework Programme/779776
dc.rights.none.fl_str_mv © The Royal Academy of Sciences, Madrid 2019
info:eu-repo/semantics/openAccess
rights_invalid_str_mv © The Royal Academy of Sciences, Madrid 2019
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
instname:Universidad Pública de Navarra
instname_str Universidad Pública de Navarra
reponame_str Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
collection Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
repository.name.fl_str_mv
repository.mail.fl_str_mv
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