Abstract
Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on one-dimensional probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from one- to two-dimensional systems is a long lasting challenging problem.
In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.
Due to space constraint, we refer the reader to [16] for the missing proofs.
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References
Aracena, J., Lamine, S.B., Mermet, M.-A., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Trans. on Systems, Man, and Cybernetics Part B 33(5), 825–834 (2003)
Bersini, H., Detours, V.: Asynchrony induces stability in cellular automata based models. In: Proceedings of Artificial Life IV, pp. 382–387. MIT Press, Cambridge (1994)
Bovier, A., Manzo, F.: Metastability in glauber dynamics in the low temperature limit: Beyond exponential asymptotics. J. Statist. Phys. 107, 757–779 (2002)
Buvel, R.L., Ingerson, T.E.: Structure in asynchronous cellular automata. Physica D 1, 59–68 (1984)
Fatès, N., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. Complex Systems 16(1), 1–27 (2005)
Fatès, N., Morvan, M., Schabanel, N., Thierry, É.: Fully asynchronous behaviour of double-quiescent elementary cellular automata. Theoretical Computer Science 362, 1–16 (2006) (An extended abstract was also published in Proc. of MFCS’2005)
Fatès, N., Regnault, D., Schabanel, N., Thierry, É.: Asynchronous behaviour of double-quiescent elementary cellular automata. In: LATIN 2006. LNCS, vol. 3887, Springer, Heidelberg (2006)
Fukś, H.: Non-deterministic density classification with diffusive probabilistic cellular automata. Phys. Rev. E 66(2) (2002)
Fukś, H.: Probabilistic cellular automata with conserved quantities. Nonlinearity 17(1), 159–173 (2004)
Goles, E., Martinez, S.: Neural and automata networks, dynamical behavior and applications. In: Maths and Applications, vol. 58, Kluwer Academic Publishers, Dordrecht (1990)
http://mathworld.wolfram.com/Outer-TotalisticCellularAutomaton.html
Lumer, E.D., Nicolis, G.: Synchronous versus asynchronous dynamics in spatially distributed systems. Physica D 71, 440–452 (1994)
Randall, D.: Mixing. In: Proc. of the Symp. on Foundations of Computer Science (FOCS), pp. 4–15 (2003)
Regnault, D.: Abrupt behaviour changes in cellular automata under asynchronous dynamics. In: Proceedings of 2nd European Conference on Complex Systems (ECCS), Oxford, UK (2006) (to appear)
Regnault, D., Schabanel, N., Thierry, É.: Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority (Full text). Preprint arXiv:0706.2479 [cs.DM] (2007)
D. Regnault, N. Schabanel, and É. Thierry. A study of stochastic 2D Minority CA: Would wearing stripes be a fatality for snob people? Research Report No ENSL-00140883, École Normale Supérieure de Lyon, 2007.
Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51, 123–143 (1999)
Tarjan, R.E.: Amortized computational complexity. SIAM Journal of Algebraic and Discrete Methods 6(2), 306–318 (1985)
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Regnault, D., Schabanel, N., Thierry, É. (2007). Progresses in the Analysis of Stochastic 2D Cellular Automata: A Study of Asynchronous 2D Minority. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_30
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DOI: https://doi.org/10.1007/978-3-540-74456-6_30
Publisher Name: Springer, Berlin, Heidelberg
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