Abstract
We study herewith the simple threshold cellular automata (CA), as perhaps the simplest broad class of CA with non-additive (that is, non-linear and non-affine) local update rules. We characterize all possible computations of the most interesting rule for such CA, namely, the Majority (MAJ) rule, both in the classical, parallel CA case, and in case of the corresponding sequential CA where the nodes update sequentially, one at a time. We compare and contrast the configuration spaces of arbitrary simple threshold automata in those two cases, and point out that some parallel threshold CA cannot be simulated by any of their sequential equivalents. We show that the temporal cycles exist only in case of (some) parallel simple threshold CA, but can never take place in sequential threshold CA. We also show that most threshold CA have very few fixed point configurations and few (if any) cycle configurations, and that, while the MAJ sequential and parallel CA may have many fixed points, nonetheless “almost all” configurations, in both parallel and sequential cases, are transient states. Finally, motivated by the contrasts between parallel and sequential simple threshold CA, we try to motivate the study of genuinely asynchronous CA.
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References
Ross Ashby, W.: Design for a Brain. Wiley, Chichester (1960)
Barrett, C., Reidys, C.: Elements of a theory of computer simulation I: sequential CA over random graphs. Applied Math. & Comput. 98(2-3) (1999)
Barrett, C., Hunt, H., Marathe, M., Ravi, S.S., Rosenkrantz, D., Stearns, R., Tosic, P.: Gardens of Eden and Fixed Points in Sequential Dynamical Systems. Discrete Math. & Theoretical Comp. Sci. Proc. AA (DM-CCG) (July 2001)
Barrett, C., Hunt III, H.B., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: Reachability problems for sequential dynamical systems with threshold functions. TCS 1-3, 41–64 (2003)
Barrett, C., Mortveit, H., Reidys, C.: Elements of a theory of computer simulation II: sequential dynamical systems. Applied Math. & Comput. 107(2-3) (2000)
Barrett, C., Mortveit, H., Reidys, C.: Elements of a theory of computer simulation III: equivalence of sequential dynamical systems. Appl. Math. & Comput. 122(3) (2001)
Garzon, M.: Models of Massive Parallelism: Analysis of Cellular Automata and Neural Networks. Springer, Heidelberg (1995)
Goles, E., Martinez, S.: Neural and Automata Networks: Dynamical Behavior and Applications. Math. & Its Applications series, vol. 58. Kluwer, Dordrecht (1990)
Goles, E., Martinez, S. (eds.): Cellular Automata and Complex Systems. Nonlinear Phenomena and Complex Systems series. Kluwer, Dordrecht (1999)
Ingerson, T.E., Buvel, R.L.: Structure in asynchronous cellular automata. Physica D: Nonlinear Phenomena 10(1-2) (January 1984)
Kauffman, S.A.: Emergent properties in random complex automata. Physica D: Nonlinear Phenomena 10(1-2) (January 1984)
Milner, R.: ACalculus of Communicating Systems. Lecture Notes Comp. Sci. Springer, Berlin (1989)
Milner, R.: Calculi for synchrony and asynchrony. Theoretical Comp. Sci., vol. 25. Elsevier, Amsterdam (1983)
Milner, R.: Communication and Concurrency. C. A. R. Hoare series ed. Prentice-Hall Int’l, Englewood Cliffs (1989)
von Neumann, J.: Theory of Self-Reproducing Automata, edited and completed by Burks, A.W. Univ. of Illinois Press, Urbana (1966)
Reynolds, J.C.: Theories of Programming Languages. Cambridge Univ. Press, Cambridge (1998)
Sethi, R.: Programming Languages: Concepts & Constructs, 2nd edn. Addison-Wesley, Reading (1996)
Sutner, K.: Computation theory of cellular automata. In: MFCS 1998 Satellite Workshop on CA, Brno, Czech Rep. (1998)
Tosic, P., Agha, G.: Concurrency vs. Sequential Interleavings in 1-D Cellular Automata. In: APDCM Workshop, Proc. IEEE IPDPS 2004, Santa Fe, New Mexico (2004)
Wolfram, S.: Twenty problems in the theory of CA. Physica Scripta 9 (1985)
Wolfram, S. (ed.): Theory and applications of CA. World Scientific, Singapore (1986)
Wolfram, S.: Cellular Automata and Complexity (collected papers). Addison-Wesley, Reading (1994)
Wolfram, S.: A New Kind of Science. Wolfram Media, Inc., Champaign (2002)
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Tosic, P.T., Agha, G.A. (2004). Characterizing Configuration Spaces of Simple Threshold Cellular Automata. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds) Cellular Automata. ACRI 2004. Lecture Notes in Computer Science, vol 3305. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30479-1_89
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DOI: https://doi.org/10.1007/978-3-540-30479-1_89
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