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Accretive Rules in Cayley P Systems

  • Jean-Louis Giavitto 
  • Olivier Michel
  • Julien Cohen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2597)

Abstract

During a discussion taking place at WMC’01, G. Paun put the question of what could be computed only by moving symbols between membranes. In this paper we provide some elements of the answer, in a setting similar to tissue Psystems, where the set of membranes is organized into a finite graph or into a Cayley graph, and using a very simple propagation process characterizing accretive growth. Our main result is to characterize the final configuration as a least fixed point and to establish two series of approximations that converge to it. All the notions introduced (Cayley graph of membranes, accretive rule and iteration) have been implemented in the MGS programming language and the two approximation series can be effectively computed in Pressburger arithmetics using the omega calculator in the case of Abelian Cayley graphs.

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References

  1. Coh96.
    A. Cohen. Structure de données régulières et analyse de flot. DEA, ENS-Lyon, June 1996.Google Scholar
  2. Fea91.
    P. Feautrier. Dataflow analysis of scalar and array references. Int. Journal of Parallel Programming, 20(1):23–53, February 1991.zbMATHCrossRefGoogle Scholar
  3. GGMP02.
    J.-L. Giavitto, C. Godin, O. Michel, and P. Prusinkiewicz. Biological Modeling in the Genomic Context, chapter “Computational Models for Integrative and Developmental Biology”. Hermes, July 2002.Google Scholar
  4. Gia99.
    J.-L. Giavitto. Scientific report for the tenure. Technical report, LRI, Université de Paris-Sud, centre d’Orsay, September 1999. Research Report 1226.Google Scholar
  5. Gia00.
    J.-L. Giavitto. A framework for the recursive definition of data structures. In Proceedings of the 2nd Imternational ACM SIGPLANConference on Principles and Practice of Declarative Programming (PPDP-00), pages 45–55. ACM Press, September 20–23 2000.Google Scholar
  6. GM02.
    J.-L. Giavitto and O. Michel. The topological structures of membrane computing. Fundamenta Informaticae, 49:107–129, 2002.MathSciNetGoogle Scholar
  7. Hey97.
    M.-C. Heydemann. Graph Symmetry, chapter “Cayley graphs and interconnection networks”, pages 167–224. Kluwer Academic Publisher, 1997.Google Scholar
  8. Joi87.
    B. Joinnault. Conception d’algorithmes et d’architecture systoliques. PhD thesis, Thèse de l’Université de Rennes I, September 1987.Google Scholar
  9. KMP+96.
    W. Kelly, V. Maslov, W. Pugh, E. Rosser, T. Shpeisman, and D. Wonnacott. The Omega calculator and library, version 1.1.0. College Park, MD 20742, 18 november 1996.Google Scholar
  10. KMW67.
    R.M. Karp, R.E. Miller, and S. Winograd. The organization of computations for uniform recurrence equations. Journal of the ACM, 14(3):563–590, July 1967.MathSciNetzbMATHCrossRefGoogle Scholar
  11. KPRS94.
    W. Kelly, W. Pugh, E. Rosser, and T. Shpeisman. Transitive closure of infinite graphs and its application. Technical Report UMIACS-TR-95-48, CS-TR-3457, Univ. of Maryland, College Park, MD 20742, 14 Aprils 1994.Google Scholar
  12. LC94.
    B. Lisper and J.-F. Collard. Extent analysis of data fields. Technical Report TRITA-IT R 94:03, Royal Institute of Technology, Sweden, January 1994.Google Scholar
  13. MVPPRP01.
    C. Martin-Vide, G. Paun, J. Pazos, and A. Rodriguez-Paton. Tissue PSystems. Technical Report TUCS tech. rep. 421, Turku Centre for Computer Science, September 2001.Google Scholar
  14. Pau99.
    G. Paun. Computing with membranes: An introduction. Bulletin of the European Association for Theoretical Computer Science, 67:139–152, February 1999.Google Scholar
  15. Pau00.
    G. Paun. Computing with membranes. Journal of Computer and System Sciences, 61:108–143, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Pau01.
    G. Paun. From cells to computers: Computing with membranes (p systems). Biosystems, 59(3):139–158, March 2001.CrossRefGoogle Scholar
  17. PSY01.
    G. Paun, Y. Sakakibara, and T. Yokomori. P systems on graphs of restricted forms. Publ. Math. Debrecen, 2001.Google Scholar
  18. Rók94.
    Z. Róka. One-way cellular automata on Cayley graphs. Theoretical Computer Science, 132(1–2):259–290, 26 September 1994.Google Scholar
  19. Rók95a.
    Z. Róka. The firing squad synchronization problem on Cayley graphs. Lecture Notes in Computer Science, 969:402–411, 1995.Google Scholar
  20. Rók95b.
    Z. Róka. Simulations between cellular automata on Cayley graphs. Lecture Notes in Computer Science, 911:483–493, 1995.Google Scholar
  21. SQ93.
    Y. Saouter and P. Quinton. Computability of recurrence equations. Theoretical Computer Science, 116(2):317–337, August 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  22. TTK95.
    R. Thomas, D. Thieffry, and M. Kaufman. Dynamical behaviours of regulatory networks-I. biological role of feedback loops and pratical use of the concept of feedback loop. Bulletin of Mathematical Biology, 57(2):247–276, 1995.zbMATHGoogle Scholar
  23. YPQ58.
    H.P. Yockey, R.P. Platzman, and H. Quastler, editors. Symposium on Information Theory in Biology. Pergamon Press, New York, London, 1958.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jean-Louis Giavitto 
    • 1
  • Olivier Michel
    • 1
  • Julien Cohen
    • 1
  1. 1.LaMI, umr 8042 du CNRSUniversité d’Évry - GENOPOLEÉvryFrance

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