On the Cost of Simulating a Parallel Boolean Automata Network by a Block-Sequential One

  • Florian Bridoux
  • Pierre Guillon
  • Kévin Perrot
  • Sylvain Sené
  • Guillaume Theyssier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


In this article we study the minimum number \(\kappa \) of additional automata that a Boolean automata network (BAN) associated with a given block-sequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call \({{\mathrm{\mathsf {NECC}}}}\) graph built from the BAN and the update schedule. We show the relation between \(\kappa \) and the chromatic number of the \({{\mathrm{\mathsf {NECC}}}}\) graph. Thanks to this \({{\mathrm{\mathsf {NECC}}}}\) graph, we bound \(\kappa \) in the worst case between n / 2 and \(2n/3+2\) (n being the size of the BAN simulated) and we conjecture that this number equals n / 2. We support this conjecture with two results: the clique number of a \({{\mathrm{\mathsf {NECC}}}}\) graph is always less than or equal to n / 2 and, for the subclass of bijective BANs, \(\kappa \) is always less than or equal to \(n/2+1\).


Boolean automata networks Intrinsic simulation Block-sequential update schedules 



This work has been partially supported by the project PACA APEX FRI.


  1. 1.
    Aracena, J.: On the robustness of update schedules in boolean networks. Biosystems 97, 1–8 (2009)CrossRefGoogle Scholar
  2. 2.
    Bruck, J., Goodman, J.W.: A generalized convergence theorem for neural networks. IEEE Trans. Inf. Theor. 34, 1089–1092 (1988)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking I: an abstract theory of bulking. Theor. Comput. Sci. 412, 3866–3880 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking II: classifications of cellular automata. Theor. Comput. Sci. 412, 3881–3905 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Demongeot, J., Elena, A., Sené, S.: Robustness in regulatory networks: a multi-disciplinary approach. Acta Biotheor. 56, 27–49 (2008)CrossRefGoogle Scholar
  6. 6.
    Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of FOCS 2012, pp. 302–310. IEEE Computer Society (2012)Google Scholar
  7. 7.
    Goles, E., Martínez, S.: Neural and Automata Networks: Dynamical Behavior and Applications. Kluwer Academic Publishers, Dordrecht (1990)CrossRefMATHGoogle Scholar
  8. 8.
    Goles, E., Matamala, M.: Computing complexity of symmetric quadratic neural networks. In: Proceedings of ICANN 1993, p. 677 (1993)Google Scholar
  9. 9.
    Goles, E., Noual, M.: Disjunctive networks and update schedules. Adv. Appl. Math. 48, 646–662 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goles, E., Salinas, L.: Comparison between parallel and serial dynamics of boolean networks. Theor. Comput. Sci. 396, 247–253 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Guillon, P.: Projective subdynamics and universal shifts. In: DMTCS Proceedings of AUTOMATA 2011, pp. 123–134 (2011)Google Scholar
  12. 12.
    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79, 2554–2558 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kauffman, S.: Gene regulation networks: a theory for their global structures and behaviors. Curr. Top. Dev. Biol. 6, 145–181 (1971). SpringerCrossRefGoogle Scholar
  14. 14.
    Kauffman, S.A.: Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lafitte, G., Weiss, M.: Universal tilings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 367–380. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-70918-3_32 CrossRefGoogle Scholar
  16. 16.
    Lafitte, G., Weiss, M.: An almost totally universal tile set. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 271–280. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02017-9_30 CrossRefGoogle Scholar
  17. 17.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. J. Math. Biophys. 5, 115–133 (1943)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Melliti, T., Regnault, D., Richard, A., Sené, S.: On the convergence of boolean automata networks without negative cycles. In: Kari, J., Kutrib, M., Malcher, A. (eds.) AUTOMATA 2013. LNCS, vol. 8155, pp. 124–138. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40867-0_9 CrossRefGoogle Scholar
  19. 19.
    Melliti, T., Regnault, D., Richard, A., Sené, S.: Asynchronous simulation of boolean networks by monotone boolean networks. In: El Yacoubi, S., Wąs, J., Bandini, S. (eds.) ACRI 2016. LNCS, vol. 9863, pp. 182–191. Springer, Cham (2016). doi: 10.1007/978-3-319-44365-2_18 CrossRefGoogle Scholar
  20. 20.
    Noual, M.: Updating automata networks. Ph.D. thesis, École Normale Supérieure de Lyon (2012)Google Scholar
  21. 21.
    Noual, M., Regnault, D., Sené, S.: About non-monotony in boolean automata networks. Theor. Comput. Sci. 504, 12–25 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ollinger, N.: Universalities in cellular automata. In: Rozenberg, G., et al. (eds.) Handbook of Natural Computing, pp. 189–229. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Orponen, P.: Computing with truly asynchronous threshold logic networks. Theor. Comput. Sci. 174, 123–136 (1997)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tchuente, M.: Sequential simulation of parallel iterations and applications. Theor. Comput. Sci. 48, 135–144 (1986)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585 (1973)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Florian Bridoux
    • 1
  • Pierre Guillon
    • 2
  • Kévin Perrot
    • 1
  • Sylvain Sené
    • 1
    • 3
  • Guillaume Theyssier
    • 2
  1. 1.Université d’Aix-Marseille, CNRS, LIFMarseilleFrance
  2. 2.Université d’Aix-Marseille, CNRS, Centrale Marseille, I2MMarseilleFrance
  3. 3.Institut rhône-alpin des systèmes complexes, IXXILyonFrance

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