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

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

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.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Pública de Navarra, Campus Arrosadia, Iruña-Pamplona, 31006, Spain

    Asier Estevan

  2. Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Ctra. de Valldemossa km. 7.5, 07122, Palma de Mallorca, Spain

    Juan-José Miñana & Oscar Valero

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  1. Asier Estevan
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  2. Juan-José Miñana
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  3. Oscar Valero
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Correspondence to Oscar Valero.

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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. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.

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Estevan, A., Miñana, JJ. & Valero, O. On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms. RACSAM 113, 3233–3252 (2019). https://doi.org/10.1007/s13398-019-00691-8

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  • DOI: https://doi.org/10.1007/s13398-019-00691-8

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