Abstract
Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this “freezing-order”.
We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.
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References
Banks, E.R.: Universality in cellular automata. In: Eleventh Annual Symposium on Switching and Automata Theory, Santa Monica, California. IEEE (1970)
Bollobás, B., Smith, P., Uzzell, A.: Monotone cellular automata in a random environment. Comb. Probab. Comput. 24(4), 687–722 (2015)
Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a Bethe lattice. J. Phys. C: Solid State Phys. 12, L31–L35 (1979)
Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking I: an abstract theory of bulking. Theor. Comput. Sci. 412(30), 3866–3880 (2011)
Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking II: classifications of cellular automata. Theor. Comput. Sci. 412(30), 3881–3905 (2011)
Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: FOCS 2012 Proceedings, pp. 302–310 (2012)
Fuentes, M.A., Kuperman, M.N.: Cellular automata and epidemiological models with spatial dependence. Physica A: Stat. Mech. Appl. 267(3–4), 471–486 (1999)
Goles, E., Montealegre-Barba, P., Todinca, I.: The complexity of the bootstraping percolation and other problems. Theor. Comput. Sci. 504, 73–82 (2013)
Goles, E., Ollinger, N., Theyssier, G.: Introducing freezing cellular automata. In: Kari, J., Törmä, I., Szabados, M. (eds.) Exploratory Papers of Cellular Automata and Discrete Complex Systems (AUTOMATA 2015), pp. 65–73 (2015)
Gravner, J., Griffeath, D.: Cellular automaton growth on Z2: theorems, examples, and problems. Adv. Appl. Math. 21(2), 241–304 (1998)
Griffeath, D., Moore, C.: Life without death is P-complete. Complex Syst. 10, 437–448 (1996)
Holland, J.H.: A universal computer capable of executing an arbitrary number of subprograms simultaneously. In: Bukrs, A.W. (ed.) Essays on Cellular Automata, pp. 264–276. University of Illinois Press, Champaign (1970)
Lind, D.A., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, New York (1995)
Meunier, P.-E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: SODA 2014 Proceedings, pp. 752–771 (2014)
Von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Champaign (1966)
Ollinger, N.: Universalities in cellular automata. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds.) Handbook of Natural Computing, pp. 189–229. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-540-92910-9_6
Ollinger, N., Richard, G.: Four states are enough!. Theor. Comput. Sci. 412(1–2), 22–32 (2011)
Thatcher, J.W.: Universality in the von neumman cellular model. In: Bukrs, A.W. (ed.) Essays on Cellular Automata, pp. 132–186. University of Illinois Press, Champaign (1970)
Ulam, S.M.: On some mathematical problems connected with patterns of growth of figures. In: Bukrs, A.W. (ed.) Essays on Cellular Automata, pp. 219–231. University of Illinois Press, Champaign (1970)
Vollmar, R.: On cellular automata with a finite number of state changes. In: Knödel, W., Schneider, H.-J. (eds.) Parallel Processes and Related Automata. Computing Supplementum, vol. 3, pp. 181–191. Springer, Vienna (1981). https://doi.org/10.1007/978-3-7091-8596-4_13
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Becker, F., Maldonado, D., Ollinger, N., Theyssier, G. (2018). Universality in Freezing Cellular Automata. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_5
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DOI: https://doi.org/10.1007/978-3-319-94418-0_5
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