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 algorithms Estevan Muguerza, Asier Miñana, Juan José Valero, Óscar Partial order Quasi-metric Fixed point Kleene Asymptotic complexity Recurrence equation 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. 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. Springer https://hdl.handle.net/2454/36069
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
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
topic Partial order
Quasi-metric
Fixed point
Kleene
Asymptotic complexity
Recurrence equation
topic_facet 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
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url https://hdl.handle.net/2454/36069
eu_rights_str_mv openAccess
publisher Springer
institution Universidad Pública de Navarra
collection Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
reponame_str Academica-e. Repositorio Institucional de la Universidad Pública de Navarra
instname_str Universidad Pública de Navarra
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publishDateSort 2019
author_browse Estevan Muguerza, Asier
Miñana, Juan José
Valero, Óscar
publisherStr Springer
score 6,924472

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