Abstract
This paper is concerned with the study of square Boolean four-neighbor peripheral cellular automata with boundary conditions fixed as zero. It is first shown that, due to plane reflection symmetry transformations, the number of dynamically nonequivalent such automata is equal to 9 616. The dynamics of the two homogeneous configurations are studied and it is shown that, in contrast with what happens in the case of periodic boundary conditions, the homogeneous configuration consisting only of ones can have various interesting dynamics, including the one typically observed for automata in Wolfram’s Class III. Finally, an example of a rule where there is coexistence between a homogeneous final state and other dynamics is presented.
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References
Binder, P.M.: Proof that a cellular automaton has a period-two global attractor. J. Phys. A: Math. Gen. 24(7), 1677–1679 (1991)
Binder, P.M.: Topological classification of cellular automata. J. Phys. A: Math. Gen. 34, L31–L34 (1991)
Binder, P.M., Twining, C.J., Sherrington, D.: Phase-space study of bistable cellular automata. Complex Systems 5(127), 127–137 (1991)
Chopard, B., Droz, M.: Cellular Automata Modeling of Physical Systems. Cambridge University Press (1998)
Chua, L.O., Guan, J., Sbitnev, V.I., Shin, J.: A nonlinear dynamics perspective of Wolfram’s New Kind of Science. Part VII: Isles of Eden. Int. J. Bifurcat. Chaos 17, 2839–3012 (2007)
Chua, L.O., Sbitnev, V.I., Yoon, S.: A nonlinear dynamics perspective of Wolfram’s New Kind of Science. Part III: Predicting the unpredictable. Int. J. Bifurcat. Chaos 14(1), 3689–3820 (2004)
Chua, L.O., Sbitnev, V.I., Yoon, S.: A nonlinear dynamics perspective of Wolfram’s New Kind of Science. Part IV: From Bernoulli shift to 1/f spectrum. Int. J. Bifurcat. Chaos 15(4), 1045–1183 (2005)
Dihidar, K., Choudhury, P.P.: Matrix algebraic formulae concerning some exceptional rules of two-dimensional cellular automata. Inf. Sci. 165, 91–101 (2004)
Freitas, J.A., Severino, R.: 2D elementray cellular automata with four neighbors. Int. J. Bifurcat. Chaos 23, 1350060 (2013)
Guan, J., Shen, S., Tang, C., Chen, F.: Extending Chua’s global equivalence theorem on Wolfram’s New Kind of Science. Int. J. Bifurcat. Chaos 17, 4245–4259 (2007)
Ilachinski, A.: Cellular Automata: A Discrete Universe. World Scientific, Singapore (2001)
Kawahara, Y., Kumamoto, S., Mizoguchi, Y., Nohmi, M., Ohtsuka, H., Shoudai, T.: Period lengths of cellular automata on square lattices with rule 90. J. Math. Phys. 36, 1435–1456 (1995)
Kerszberg, M., Mukamel, D.: Dynamics of simple computer networks. J. Stat. Phys. 51, 777–795 (1988)
Khan, A., Choudhury, P., Dihidar, K., Mitra, S., Sarkar, P.: VLSI architecture of a cellular automata machine. Comput. Math. Appl. 33(5), 79–94 (1997)
Li, W., Packard, N.: The structure of the elementary cellular automata rule. Complex Systems 4, 281–297 (1990)
Manna, S., Stauffer, D.: Systematics of transitions of square-lattice cellular automata. Physica A 162(2), 176–186 (1990)
Manneville, P., Boccara, N., Vichniac, G.Y., Bidaux, R. (eds.): Cellular Automata and Modeling of Complex Physical Systems. Springer, Berlin (1989)
Nayak, D.R., Sahu, S.K., Mohammed, J.: A cellular automata based optimal edge detection technique using twenty-five neighborhood model. Int. J.l Comput. Appl. 84(10), 27–33 (2013)
von Neumann, J.: The general and logical theory of automata. In: Jeffress, L.A. (ed.) Cerebral Mechanisms in Behavior - The Hixon Symposium, pp. 1–31. John Wiley and Sons, New York (September1948, 1951)
von Neumann, J.: Theory of Self-Reproducing Automata (Edited by A. Burks). Univeristy of Illinois Press (1966)
Pitsianis, N., Tsalides, P., Bleris, G.L., Thanailakis, A., Card, H.C.: Deterministic one-dimensional cellular automata. J. Stat. Phys. 56, 99–112 (1989)
Pries, W., Thanailakis, A., Card, C.: Group properties of cellular automata and VLSI applications. IEEE Trans. Comput. 35(12), 1013–1024 (1986)
Schweitzer, F.: Applications of cellular automata in complex systems. Adv. Complex Syst. 5(2), 101–337 (2002)
Twining, C.J.: The limiting behavior of non-cylindrical elementary cellular automata. Complex Systems 6, 417–432 (1992)
Twining, C., Binder, P.M.: Enumeration of limit cycles in noncylindrical cellular automata. J. Stat.l Phys. 66(1-2), 385–401 (1992)
Ulam, S.: Random processes and transformations. In: Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, August 30–September 6 1950, vol. 2, pp. 264–275 (1952)
Walker, C., Aadryan, A.: Amount of computation preceding externally detectable steady-state behavior in a class of complex systems. Int. J. Biomed. Comput. 2(2), 85–94 (1971)
Wolfram, S.: Computation theory of cellular automata. Commun. Math. Phys. 96, 15–57 (1984)
Wolfram, S.: Universality and complexity in cellular automata. Physica D 10, 1–35 (1984)
Wolfram, S. (ed.): Theory and Applications of Cellular Automata. World Scientific Press, Singapore (1986)
Wolfram, S.: A New Kind of Science. Wofram Media, Inc. (2002)
Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 5686–5697 (1983)
Wuensche, A., Lesser, M.: The Global Dynamics of Cellular Automata. Santa Fe Institute Studies in the Sciences of Complexity. Addison-Wesley (1992)
Zhai, Y., Yi, Z., Deng, P.-M.: On behavior of two-dimensional cellular automata with an exceptional rule. Inf. Sci. 179, 613–622 (2009)
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Severino, R., Soares, M.J., Athayde, M.E. (2014). Homogeneous Dynamics for Square Boolean Automata with Null Boundary Conditions. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2014. ICCSA 2014. Lecture Notes in Computer Science, vol 8579. Springer, Cham. https://doi.org/10.1007/978-3-319-09144-0_3
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DOI: https://doi.org/10.1007/978-3-319-09144-0_3
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