Advertisement

A Relational View of Recurrence and Attractors in State Transition Dynamics

  • Giuseppe Scollo
  • Giuditta Franco
  • Vincenzo Manca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4136)

Abstract

The classical dynamics concepts of recurrence and attractor are analysed in the basic mathematical setting of state transition systems, where both time and space are discrete, and no structure is assumed on the state space besides a binary transition relation. This framework proves useful to the dynamical analysis of computations and biomolecular processes. Here a relational formulation of this framework is presented, where the concepts of attractor and recurrence surface in two variants, respectively relating to the two fundamental modalities. A strong link between recurrence and both existence and extent of attractors, in either variant, is established by a novel characterization theorem.

Keywords

Binary Relation Individual State Relational Formulation Relation Algebra Relational View 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ashby, W.R.: An Introduction to Cybernetics. Chapman and Hall, Boca Raton (1956)MATHGoogle Scholar
  2. 2.
    Backhouse, R., van der Woude, J.: Demonic Operators and Monotype Factors. Mathematical Structures in Computer Science 3(4), 417–433 (1993)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bonanno, C., Manca, V.: Discrete dynamics in biological models. Romanian Journal of Information Science and Technology 1-2(5), 45–67 (2002)Google Scholar
  4. 4.
    Bianco, L., Fontana, F., Franco, G., Manca, V.: P Systems for Biological Dynamics. In: Ciobanu, G., Păun, Gh., Perez-Jimenez, M.J. (eds.) Applications of Membrane Computing. Natural Computing Series, pp. 81–126. Springer, Heidelberg (2006)Google Scholar
  5. 5.
    Devaney, R.L.: Introduction to chaotic dynamical systems. Addison-Wesley, Reading (1989)MATHGoogle Scholar
  6. 6.
    Franco, G.: Biomolecular Computing – Combinatorial Algorithms and Laboratory Experiments, PhD thesis, University of Verona, Italy (2006)Google Scholar
  7. 7.
    Kauffman, S.: Investigations. Oxford University Press, Oxford (2000)Google Scholar
  8. 8.
    Kůrka, P.: Topological and Symbolic Dynamics, Cours Spécialisés. Société Mathématique de France 11 (2003)Google Scholar
  9. 9.
    Manca, V., Franco, G., Scollo, G.: State transition dynamics: basic concepts and molecular computing perspectives. In: Gheorghe, M. (ed.) Molecular Computational Models: Unconventional Approaches, Idea Group, Hershey, PA, USA, pp. 32–55 (2005)Google Scholar
  10. 10.
    Manca, V., Bianco, L.: Biological Networks in Metabolic P Systems (submitted, 2006)Google Scholar
  11. 11.
    Manca, V., Bianco, L., Fontana, F.: Evolution and oscillation in P systems: Applications to biological phenomena. In: Mauri, G., Păun, Gh., Jesús Pérez-Jímenez, M., Rozenberg, G., Salomaa, A. (eds.) WMC 2004. LNCS, vol. 3365, pp. 63–84. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Păun, Gh.: Computing with membranes. J. Comput. System Sci. 61(1), 108–143 (2000)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Păun, Gh.: Membrane Computing. An Introduction. Springer, Heidelberg (2002)MATHGoogle Scholar
  14. 14.
    Schmidt, G., Ströhlein, T.: Relations and Graphs. Springer, Heidelberg (1993)MATHGoogle Scholar
  15. 15.
    Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6, 73–89 (1941)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Wolfram, S.: Theory and Application of Cellular Automata. Addison-Wesley, Reading (1986)Google Scholar
  17. 17.
    Wuensche, A.: Basins of Attraction in Network Dynamics: A Conceptual Framework for Biomolecular Networks. In: Schlosser, G., Wagner, G.P. (eds.) Modularity in Development and Evolution, Chicago University Press (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Giuseppe Scollo
    • 1
  • Giuditta Franco
    • 2
  • Vincenzo Manca
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  2. 2.Department of Computer ScienceUniversity of VeronaVeronaItaly

Personalised recommendations