Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-02T23:42:13.977Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stochastic Automaton Model for Interacting Systems

Published online by Cambridge University Press:  05 September 2017

P. Tautu
Get access Rights & Permissions [Opens in a new window]

Abstract

A stochastic tessellation automaton (STA) is introduced and analysed as an analogue to a stochastic lattice process, called the Markov configuration process. The STA is considered as an (infinite regular) array with interconnected Moore-type automata, each of these representing a B-object interacting with its neighbours. The objective of the paper is to examine some consequences of the analogy between an STA and the Markov configuration process. In addition, the possibility of finding a suitable stochastic grammar arising from the study of this configuration model is briefly considered.

Type
Part IX — Biomathematics and Epidemiology
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 403 - 415
Copyright
Copyright © 1975 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aho, A. V. and Ullman, J. D. (1968) The theory of languages. Math. Systems Theory 2, 97–125.CrossRefGoogle Scholar
[2] Amoroso, S. and Guilfoyle, R. (1972) Some comments on neighborhood size for tessellation automata. Information and Control 21, 48–55.Google Scholar
[3] Arbib, M. A. (1967) Automata theory and development. J. Theor. Biol. 14, 131–156.Google Scholar
[4] Arbib, M. A. (1969) Self-reproducing automata. Some implications for theoretical biology. Towards a Theoretical Biology 2, 204–226. Edinburgh University Press.Google Scholar
[5] Bacon, G. C. (1964) The decomposition of stochastic automata. Information and Control 7, 320–339.Google Scholar
[6] Bartlett, M.S. (1967) Inference and stochastic processes. J. R. Statist. Soc. A 130, 457–477.Google Scholar
[7] Bartlett, M. S. (1968) A further note on nearest neighbour models. J. R. Statist. Soc. A 131, 579–580.Google Scholar
[8] Bartlett, M.S. (1971a) Physical nearest-neighbour models and non-linear time series. J. Appl. Prob. 8, 222–232.Google Scholar
[9] Bartlett, M.S. (197lb) Two-dimensional nearest-neighbour systems and their ecological applications. Statistical Ecology 1, 179–194. Penn. State University Press.Google Scholar
[10] Besag, J.E. (1972) Nearest-neighbour systems and the auto-logistic model for binary data. J. R. Statist. Soc. B 34, 75–83.Google Scholar
[11] Brook, D. (1964) On the distinction between the conditional probability and the joint probability approaches in specification of nearest-neighbour systems. Biometrika 51, 481–483.Google Scholar
[12] Burks, A.W. (1963) Toward a theory of automata based on more realistic primitive elements. Inform. Proc. 1962, Proc. IFIP Congr. pp. 379–385. (Reprinted in [13], pp. 84–102.)Google Scholar
[13] Burks, A.W., (ed.) (1970) Essays on Cellular Automata. University of Illinois Press, Urbana.Google Scholar
[14] Carlyle, J.W. (1963) Reduced forms of stochastic sequential machines. J. Math. Anal. Appl. 7, 167–175.Google Scholar
[15] Chung, K.L. (1967) Markov Chains with Stationary Transition Probabilities. 2nd ed. Springer, Berlin.Google Scholar
[16] Clifford, P. and Sudbury, A. (1973) A model for spatial conflict. Biometrika 60, 581–588.Google Scholar
[17] Codd, E.F. (1968) Cellular Automata. Academic Press, New York.Google Scholar
[18] Cohen, D. (1967) Computer simulation of biological pattern generation processes. Nature 216, 246–248.Google Scholar
[19] Coxeter, H.S.M. (1963) Regular Polytopes. 2nd ed. Macmillan, New York.Google Scholar
[20] Davis, A.S. (1961) Markov chains as random input automata. Amer. Math. Monthly 68, 264–267.Google Scholar
[21] Downham, D.Y. and Morgan, R.K.B. (1973) Growth of abnormal cells. Nature 242, 528–530.Google Scholar
[22] Eden, M. (1961) A two-dimensional growth process. Proc. Fourth Berkeley Symp. Math. Statist. Prob. 4, 223–239.Google Scholar
[23] Gelenbe, S.E. (1970) On the loop-free decomposition of stochastic finite-state systems. Information and Control 17, 474–484.Google Scholar
[24] Gordon, R. (1966) On stochastic growth and form. Proc. Nat. Acad. Sci. USA 56, 1497–1504.Google Scholar
[25] Grenander, U. (1970) A unified approach to pattern analysis Adv. in Computers 10, 175–216.CrossRefGoogle Scholar
[26] Hammersley, J.M. (1972) Stochastic models for the distribution of particles in space. Adv. Appl. Prob. Suppl. 47–68.Google Scholar
[27] Harris, T.E. (1972) Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Math. 9, 66–89.Google Scholar
[28] Hartmanis, J. (1962) Loop-free structure of sequential machines. Information and Control 5, 25–43.Google Scholar
[29] Hartmanis, J. and Stearns, R.E. (1966) Algebraic Structure Theory of Sequential Machines. Prentice-Hall, Englewood Cliffs.Google Scholar
[30] Herman, G.T. (1969) Computing ability of a developmental model for filamentous organisms. J. Theor. Biol. 25, 421–435.Google Scholar
[31] Holley, R. (1970) A class of interactions in an infinite particle system. Advances in Math. 5, 291–309.Google Scholar
[32] Kameda, T. (1968) Generalized transition matrix of a sequential machine and its applications. Information and Control 12, 259–275.Google Scholar
[33] Kitagawa, T. (1974) Cell space approaches in biomathematics. Math. Biosci. 19, 27–71.Google Scholar
[34] Knast, R. (1969) Continuous-time probabilistic automata. Information and Control 15, 335–352.Google Scholar
[35] Knast, R. (1970) Representability of nonregular languages in finite probabilistic automata. Information and Control 16, 285–302.Google Scholar
[36] Knast, R. (1972) Finite-state probabilistic languages. Information and Control 21, 148–170.Google Scholar
[37] Liggett, T.M. (1972) Existence theorems for infinite particle systems. Trans. Amer. Math. Soc. 165, 471–481.Google Scholar
[38] Lindenmayer, A. (1968) Mathematical models for cellular interactions in development. I, II. J. Theor. Biol. 18, 280–299; 300–315.Google Scholar
[39] Lindenmayer, A. and Rozenberg, G. (1972) Developmental systems and languages. Proc. 4th ACM Symp. Theory Computing, pp. 214–221.Google Scholar
[40] Lofgren, L. (1963) Self-repair as a computability concept in the theory of automata. Proc. Symp. Mathematical Theory of Automata, pp. 205–222. Polytechnic Press.Google Scholar
[41] Mealy, G.H. (1955) A method for synthesizing sequential circuits. Bell System Tech. J. 34, 1045–1079.Google Scholar
[42] Moore, E.F. (1956) Gedanken-experiments on sequential machines. Automata Studies. Ann. Math. Studies 34, 129–153. Princeton University Press.Google Scholar
[43] Moore, E.F. (1962) Machine models of self-reproduction. Proc. Symp. Appl. Math. 14, 17–33. (Reprinted in [13], 187–203.) Google Scholar
[44] Morgan, R.W. and Welsh, D.J.A. (1965) A two-dimensional Poisson growth process. J. R. Statist. Soc. B 27, 497–504.Google Scholar
[45] Onicescu, O. and Guiasu, S. (1965) Finite abstract random automata. Z. Wahrscheinlichkeitsth. 3, 279–285.Google Scholar
[46] Page, C.V. (1966) Equivalences between probabilistic and deterministic sequential machines. Information and Control 9, 469–520.Google Scholar
[47] Paz, A. (1971) Introduction to Probabilistic Automata. Academic Press, New York.Google Scholar
[48] Richardson, D. (1973) Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar
[49] Rosenkrantz, D.J. (1969) Programmed grammars and classes of formal languages, J. Assoc. Comput. Mach. 16, 107–131.Google Scholar
[50] Salomaa, A. (1968) On events represented by probabilistic automata of different types. Canad. J. Math. 20, 242–251.Google Scholar
[51] Salomaa, A. (1973a) On exponential growth in Lindenmayer systems. Indag. Math. 35, 23–30.Google Scholar
[52] Salomaa, A. (1973b) Formal Languages. Academic Press, New York.Google Scholar
[53] Santos, E.S. (1972) Probabilistic grammars and automata. Information and Control 21, 27–47.Google Scholar
[54] Schürger, K. and Tautu, P. (1975) Markov configuration processes on a lattice. Rev. Roumaine Math. Pures Appl. To appear.Google Scholar
[55] Shinahr, I. (1974) Two-and three-dimensional firing-squad synchronization problem. Information and Control 24, 163–180.Google Scholar
[56] Smith, A.R. III (1971) Cellular automata complexity trade-offs. Information and Control 18, 466–482.Google Scholar
[57] Spitzer, F. (1969) Random processes defined through the interaction of an infinite particle system. Lecture Notes in Math. 89, 201–223. Springer, Berlin.Google Scholar
[58] Spitzer, F. (1970) Interaction of Markov processes. Advances in Math. 5, 246–290.Google Scholar
[59] Spitzer, F. (1971) Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78, 142–154.Google Scholar
[60] Starke, P. H. and Thiele, H. (1970) On asynchronous stochastic automata. Information and Control 17, 265–293.Google Scholar
[61] Swain, P. H. and Fu, K. S. (1972) Stochastic programmed grammars for syntactic pattern recognition. Pattern Recognition 4, 83–100.CrossRefGoogle Scholar
[62] Tautu, P. (1973) Random systems of locally interacting cells. 3rd Conf. Stoch. Proc. Appl. Sheffield. (Abstract published in Adv. Appl. Prob. 6, 237.)Google Scholar
[63] Turakainen, P. (1968) On stochastic languages. Information and Control 12, 304–313.Google Scholar
[64] Ulam, S. (1952) Random processes and transformations. Proc. Intern. Congr. Mathematicians, 2, 264–275.Google Scholar
[65] Van Dalen, D. (1971) A note on some systems of Lindenmayer. Math. Systems Theory 5, 128–140.Google Scholar
[66] Vitányi, P. B. M. (1973a) Sexually reproducing cellular automata. Math. Biosci. 18, 23–54.Google Scholar
[67] Vitányi, P. B. M. (1973b) Structure of growth in Lindenmayer systems. Indag. Math. 35, 257–263.Google Scholar
[68] Whittle, P. (1963) Stochastic processes in several dimensions. Bull. Inst. Internat. Statist. 40, 974–994.Google Scholar
[69] Williams, T. and Bjerknes, R. (1972) A stochastic model for abnormal clone spread through epitelial basal layer. Nature 236, 19–21.Google Scholar
[70] Yamada, H. and Amoroso, S. (1969) Tessellation automata. Information and Control 14, 229–317.Google Scholar
[71] Yamada, H. and Amoroso, S. (1971) Structural and behavioral equivalences of tessellation automata. Information and Control 18, 1–31.Google Scholar