Skip to main content
Springer Nature Link
Log in

The Banach Contraction Principle in Fuzzy Quasi-metric Spaces and in Product Complexity Spaces: Two Approaches to Study the Cost of Algorithms with a Finite System of Recurrence Equations

  • Conference paper
Computational Intelligence (IJCCI 2010)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 399))

Included in the following conference series:

Abstract

Considering recursiveness as a unifying theory for algorithm related problems, we take advantage of algorithms formulation in terms of recurrence equations to show the existence and uniqueness of solution for the recurrence equations associated to a kind of algorithms defined as a finite system of procedures by applying the Banach contraction principle both in a suitable product of fuzzy quasi-metrics defined on the domain of words and in the product quasi-metric space of complexity spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Atkinson, M.D.: The Complexity of Algorithms. In: Computing Tomorrow: Future Research Directions in Computer Science, pp. 1–20. Cambridge, Univ. Press, New York (1996)

    Google Scholar 

  2. Brassard, G., Bratley, P.: Fundamentals of Algorithms. Prentice Hall (1996)

    MATH  Google Scholar 

  3. Castro-Company, F., Romaguera, S., Tirado, P.: An Application of the Banach Contraction Principle on the Product of Complexity Spaces to the Study of Certain Algorithms with Two Recurrence Procedures. In: Vigo-Aguiar, J. (ed.) Proceedings of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2010), Almería, Spain, vol. 5, pp. 978–984 (2010) ISBN 978-84-613-5510-5

    Google Scholar 

  4. Castro-Company, F., Romaguera, S., Tirado, P.: Application of the Banach Fixed Point Theorem on Fuzzy Quasi-Metric Spaces to Study the Cost of Algorithms with Two Recurrence Equations. In: IJCCI 2010 2nd International Joint Conference on Computational Intelligence (2010) ISBN 978-989-8425-32-4

    Google Scholar 

  5. Cho, Y.J., Grabiec, M., Radu, V.: On non Symmetric Topological and Probabilistic Structures. Nova Sci. Publ. Inc., New York (2006)

    MATH  Google Scholar 

  6. Cho, Y.J., Grabiec, M., Taleshian, A.A.: Cartesian product of PQM-spaces. J. Nonlinear Sci. Appl. 2, 60–70 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cormen, T.H., Leiserson, C.E., Stein, C., Rivest, R.L.: Introduction to Algorithms, 3rd edn. MIT Press (2001)

    Google Scholar 

  8. Flajolet, P.: Analytic Analysis of Algorithms. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 186–210. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  9. García-Raffi, L.M., Romaguera, S., Schellekens, M.: Applications of the Complexity Space to the General Probabilistic Divide and Conquer Algorithms. J. Math. Anal. Appl. 348, 346–355 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. George, A., Veeramani, P.: On Some Results in Fuzzy Metric Spaces. Fuzzy Sets and Systems 64, 395–399 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grabiec, M.: Fixed Points in Fuzzy Metric Spaces. Fuzzy Sets and Systems 27, 385–389 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gregori, V., Romaguera, S.: Fuzzy Quasi-Metric Spaces. Appl. Gen. Topology 5, 129–136 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gregori, V., Sapena, A.: On Fixed Point Theorems in Fuzzy Metric Spaces. Fuzzy Sets and Systems 125, 245–253 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kramosil, I., Michalek, J.: Fuzzy Metrics and Statistical Metric Spaces. Kybernetika 11, 326–334 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Künzi, H.P.A.: Nonsymmetric Topology. In: Proceedings of the Colloquium on Topology. Szekszárd, Colloq. Math. Soc. János Bolyai Math. Studies, Hungary, vol 4, pp. 303–338 (1995)

    Google Scholar 

  16. Matthews, S.G.: Partial Metric Topology. In: Proceedings of the 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci., vol. 728, pp. 183–197 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Rodríguez-López, J., Romaguera, S., Valero, O.: Denotational Semantics for Programming Languages, Balanced Quasi-Metrics and Fixed Points. Internat. J. Comput. Math. 85, 623–630 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Romaguera, S., Schellekens, M.: Partial Metric Monoids and Semivaluation Spaces. Topology Appl. 153, 948–962 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Romaguera, S., Sapena, A., Tirado, P.: The Banach Fixed Point Theorem in Fuzzy Quasi-Metric Spaces with Application to the Domain of Words. Topology Appl. 154, 2196–2203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Romaguera, S., Schellekens, M.: Quasi-Metric Properties of Complexity Spaces. Topology Appl. 98, 311–322 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Romaguera, S., Schellekens, M., Tirado, P., Valero, O.: Contraction Maps on Complexity Spaces and ExpoDC Algorithms. In: Proceedings of the International Conference of Computational Methods in Sciences and Engineering ICCMSE 2007, AIP Conference Proceedings, vol. 963, pp. 1343–1346 (2007)

    Google Scholar 

  22. Romaguera, S., Tirado, P.: Contraction Maps on IFQM-Spaces with Application to Recurrence Equations of Quicksort. Electronic Notes in Theoret. Comput. Sci. 225, 269–279 (2009)

    Article  MATH  Google Scholar 

  23. Saadati, R., Vaezpour, S.M., Cho, Y.J.: Quicksort Algorithm: Application of a Fixed Point Theorem in Intuitionistic Fuzzy Quasi-Metric Spaces at a Domain of Words. J. Comput. Appl. Math. 228, 219–225 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schellekens, M.: The Smyth Completion: a Common Foundation for Denotational Semantics and Complexity Analysis. Electronic Notes Theoret. Comput. Sci. 1, 535–556 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schellekens, M.: The Correspondence between Partial Metrics and Semivaluations. Theoret. Comput. Sci. 315, 135–149 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  27. Sehgal, V.M., Bharucha-Reid, A.T.: Fixed Points of Contraction Mappings on PM-spaces. Math. Systems Theory 6, 97–100 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  28. Smyth, M.B.: Quasi-uniformities: Reconciling Domains with Metric Spaces. In: Main, M.G., Mislove, M.W., Melton, A.C., Schmidt, D. (eds.) MFPS 1987. LNCS, vol. 298, pp. 236–253. Springer, Heidelberg (1988)

    Chapter  Google Scholar 

  29. Vasuki, R., Veeramani, P.: Fixed Point Theorems and Cauchy Sequences in Fuzzy Metric Spaces. Fuzzy Sets and Systems 135, 415–417 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Francisco Castro-Company, Salvador Romaguera & Pedro Tirado

Authors
  1. Francisco Castro-Company
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salvador Romaguera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pedro Tirado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Castro-Company .

Editor information

Editors and Affiliations

  1. , Images, Signals and Intelligence, University Paris-Est Creteil (UPEC), LISSI EA 3956, Paris, 77127, France

    Kurosh Madani

  2. , Departamento de Engenharia Informatica, University of Coimbra, Polo II - Pinhal de Marrocos, Coimbra, 3030, Portugal

    António Dourado Correia

  3. Systems and Robotics Institute, Evolutionary Systems and Biomedical, Instituto Superior Tecnico IST, Av. Rovisco Pais, Lisboa, 1049-001, Portugal

    Agostinho Rosa

  4. INSTICC, Polytechnic Institute of Setúbal, Setubal, 2910-595, Portugal

    Joaquim Filipe

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag GmbH Berlin Heidelberg

About this paper

Cite this paper

Castro-Company, F., Romaguera, S., Tirado, P. (2012). The Banach Contraction Principle in Fuzzy Quasi-metric Spaces and in Product Complexity Spaces: Two Approaches to Study the Cost of Algorithms with a Finite System of Recurrence Equations. In: Madani, K., Dourado Correia, A., Rosa, A., Filipe, J. (eds) Computational Intelligence. IJCCI 2010. Studies in Computational Intelligence, vol 399. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27534-0_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27534-0_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27533-3

  • Online ISBN: 978-3-642-27534-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics