Skip to main content

Part of the book series: Mathematics and Its Applications ((MAIA,volume 58))

  • 239 Accesses

Abstract

Automata Networks where introduced by Ulam, McCulloch and von Neumann to model phenomena studied in physics and biology [MP, U, VN1, VN2]. Automata Networks are discrete dynamical systems, in time and space. Roughly speaking, they are defined by a graph, either finite or infinite, where each site or vertex takes states in a finite set. Moreover, the state of a site changes according to a transition rule which takes into account only the state of its neighbors in the graph.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atlan, H., F. Fogelman-Soulie, J. Salomon, G. Weisbuch, Random Boolean Networks, Cybernetics and Systems, 12, 1981, 103.

    Article  MathSciNet  Google Scholar 

  2. Amit, J.D., H. Gutfreund, M. Sompolinsky, Spin-Glass Models of Neural Networks, Phys. Review A, 32(2), 1985, 1007–1018.

    Article  MathSciNet  Google Scholar 

  3. Allouche, J.P., Ch. Reder, Oscillations Spatio-Temporelles Engendrees par un Automate Cellulaire, Disc. Applied Maths., 1984, 215–254.

    Google Scholar 

  4. Berlekamp, E.R., J.H. Conway, R.K. Guy, Winning Ways, Ac. Press, 1985, 2, Chapter 25.

    Google Scholar 

  5. Bienenstock, E., F. Fogelman-Soulie, G. Weisbuch (eds), Disordered Systems and Biological Organization, NATO ASI Series F: Computer and Systems Sciences, 20, 1986.

    Google Scholar 

  6. Caianiello, E.R., Outline of a Theory of Thought-Processes and Thinking Machines, J. Theor. Biol., 2, 1961, 204–235.

    Article  MathSciNet  Google Scholar 

  7. Cosnard, M., E. Goles, Dynamique d’un Automate a Mémoire Modélisant le Fonctionement d’un Neurone, C.R.A.S., 299(10), 1984, 459–461.

    MathSciNet  MATH  Google Scholar 

  8. Codd, E.F., Cellular Automata, Acad. Press, 1988.

    Google Scholar 

  9. Culik, K., S. Yu, Undecidability of C.A. Classification Schemes, Complex Systems, 2(2), 1988, 177–190.

    MathSciNet  MATH  Google Scholar 

  10. Derrida, B., Dynamics of Automata, Spin Glasses and Neural Network Models, Preprint, Service de Physique Théorique, CEN-Saclay, France, 1987.

    Google Scholar 

  11. Demongeot, J., E. Goles, M. Tchuente (eds), Dynamical Systems and Cellular Automata, Acad. Press, 1985.

    MATH  Google Scholar 

  12. Dobrushin, R.L., V.I. Kryukov, A.L. Toom, Locally Interacting Systems and their Application in Biology, Lecture Notes in Mathematics, N-653, 1978.

    Book  MATH  Google Scholar 

  13. Derrida, B., Y. Pomeau, Random Networks, of Automata: A simple Annealed Approximation, Europhysics Letters, 1(2), 1986, 45–49.

    Article  Google Scholar 

  14. Derrida, B., G. Weisbuch, Evolution of Overlaps between Configurations in Random Boolean Networks, J. Physique, 47, 1980, 1297–1303.

    Google Scholar 

  15. Fogelman-Soulie, F., Contributions a une Theorie du Calcul sur Reseaux, Thesis, IMAG, Grenoble, France, 1985.

    Google Scholar 

  16. Fogelman-Soulie, F., E. Goles, G. Weisbuch, Specific Roles of the Different Boolean Mappings in Random Networks, Bull. Math. Biol, 44(5), 1982, 715–730.

    Article  MathSciNet  MATH  Google Scholar 

  17. Fogelman-Soulié, F., E. Goles, S. Martínez, C. Mejía, Energy Functions in Neural Networks with Continuous Local Functions, Complex Systems, 3, 1989, 269–293.

    MathSciNet  MATH  Google Scholar 

  18. Frisch, U., B. Hasslacher, S. Orszag, S. Wolfram, Proc. of “Workshop on Large Nonlinear Systems” 1986, Complex Systems, 1(4), 1987.

    Google Scholar 

  19. Fogelman- Soulie, F., Y. Robert, M. Tchuente (eds), Automata Networks in Computer Science; Theory and Applications, Nonlinear Science Series, Manchester Univ. Press, 1987.

    Google Scholar 

  20. Fogelman-Soulie, F., G. Weisbuch, Random Iterations of Threshold Networks and Associative Memory, SIAM J. on Computing, 16, 1987, 203–220.

    Article  MathSciNet  MATH  Google Scholar 

  21. Goles, E., Comportement Dynamique de Reseaux d’Automates, Thesis, IMAG, Grenoble, 1985.

    Google Scholar 

  22. Galperin, G.A., One-dimensional Automata Networks with Monotonie Local Interactions, Problemy Peredachi Informatsii, 12(4), 1976, 74–87.

    MathSciNet  Google Scholar 

  23. Galperin, G.A., One-Dimensional Monotonie Tesselations with Memory in Locally Interacting Systems and their Application in Biology, R.L. Dobrushin et al (eds), Lecture Notes in Mathematics, N-653, 1978, 56–71.

    Chapter  Google Scholar 

  24. Goles, E., F. Fogelman-Soulie, D. Pellagrin,. The Energy as a Tool for the Study of Threshold Networks, Disc. App. Math., 12, 1985, 261–277.

    Article  MATH  Google Scholar 

  25. Greenberg, J.M., C. Greene, S.P. Hastings A Combinatorial Problem Arising in the Study of Reaction-Diffusion Equations, SIAM J. Algebraic and Discrete Meths, 1, 1980, 34–42.

    Article  MathSciNet  MATH  Google Scholar 

  26. Goles, E., S. Martínez, Properties of Positive Functions and the Dynamics of Associated Automata Networks, Discrete Appl. Math. 18, 1987, 39–46.

    Article  MathSciNet  MATH  Google Scholar 

  27. Goles, E., S. Martínez, The One-Site Distributions of Gibbs States on Bethe Lattice are Probability Vectors of Period ≤ 2 for a Nonlinear Transformation, J. Stat. Physics, 52(1/2), 1988, 267–285.

    Article  MATH  Google Scholar 

  28. Goles, E., S. Martínez (eds), Proc. Congrés Franco-Chilien en Math. Appliquées, 1986, in Revista de Matemáticas Aplicadas, 9(2), 1988

    Google Scholar 

  29. Goles, E., J. Olivos, Compartement Itératif des Fonctions a Multiseuil, Information and Control, 45(3), 1980, 300–313.

    Article  MathSciNet  MATH  Google Scholar 

  30. Goles, E., J. Olivos, Compartement Periodique des Fonctions a Seuil Binaires et Applications, Disc. Appl. Maths., 3, 1981, 93–105.

    Article  MathSciNet  MATH  Google Scholar 

  31. Goles, E., J. Olivos, Periodic Behaviour of Generalized Threshold Functions, Disc. Maths., 30, 1980, 187–189.

    Article  MathSciNet  MATH  Google Scholar 

  32. Goles, E., A.M. Odlyzko, Decreasing Energy Functions and Lengths of Transients for some Lengths of Transients for Some Cellular Automata, Complex Systems, 2(5), 1988, 501–507.

    MathSciNet  MATH  Google Scholar 

  33. Hopfield, J.H., Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proc. Nat. Acad. Sei., UAS, 79, 1982, 2554–2558.

    Article  MathSciNet  Google Scholar 

  34. Hardy, J., O. de Pazzis, Y. Pomeau, Time Evolution of a Two-Dimensional Model System: Invariant States and Time Correlation Functions, J. Math. Phy., 14, 1973, 174.

    Article  Google Scholar 

  35. Hardy, J., O. de Pazzis, Y. Pomeau, Molecular Dynamics of a Classical Lattice Gas: Transport Properties and Time Correlation Functions, Phys. Rev. A. 13, 1976, 1949.

    Article  Google Scholar 

  36. Hurd, L.P., Formal Language Characterizations of Cellular Automaton Limit Sets, Complex Systems, 1, 1987, 69–80.

    MathSciNet  MATH  Google Scholar 

  37. Kleene, S.C., Representation of Events in Nerve Nets and Finite Automata in Automata Studies, C.E. Shannon and J. McCarthy (eds), Annals of Mathematics Studies, 34, Princeton Univ. Press, 1956, 3–41.

    Google Scholar 

  38. Kauffman, S.A., Behaviour of Randomly Constructed Genetic Nets in Towards a Theoretical Biology, C.H. Waddington (ed), 3, Edinburgh Univ. Press, 1970, 18–46.

    Google Scholar 

  39. Kitagawa, T., Cell Space Approaches in Biomathematics, Math. Biosciences, 19, 1974, 27–71.

    Article  MathSciNet  MATH  Google Scholar 

  40. Kobuchi, Y., Signal Propagation in 2-Dimensional Threshold Cellular Space, J. of Math. Biol., 3, 1976, 297–312.

    Article  MathSciNet  MATH  Google Scholar 

  41. Kindermann, R., J.L. Sneel, Markov Random Fields and their Applications, Series on Contemporary Mathematics, AMS, 1, 1980.

    Google Scholar 

  42. Little, W.A. Existence of Persistent States in the Brain, Math. Bios, 19, 1974, 101.

    Article  MATH  Google Scholar 

  43. Little, W.A., G.L. Shaw, Analytic Study of the Memory Storage Capacity of a Neural Network, Math. Bios, 39, 1978, 281–290.

    Article  MathSciNet  MATH  Google Scholar 

  44. Martinez, S., Cylinder Distribution of Thermodynamic Limit of Bethe Lattice, Instabilities and Non-Equilibrium Structures II, Mathematics and Its Applications, Kluwer, 1989, 117–130.

    Google Scholar 

  45. Minsky, M.L., Computation: Finite and Infinite Machines, Prentice-Hall series in Automatic Computation, 1967.

    MATH  Google Scholar 

  46. Minsky, M., S. Papert, Perceptrons, an Introduction to Computational Geometry, MIT Press, 1969.

    MATH  Google Scholar 

  47. Manneville P., N. Boccara, G. Vichniac (eds), Cellular Automata and Modeling of Complex Systems, Procc. in Physics, Springer-Verlag, 46, 1989.

    MATH  Google Scholar 

  48. McCulloch, W., W. Pitts, A Logical Calculus of the Ideas Immanent in Nervous Activity, Bull. Math. Biophysics, 5, 1943, 115–133.

    Article  MathSciNet  MATH  Google Scholar 

  49. Mezard, M., G. Parisi, M.A. Virasoro (eds), Spin Glass Theory and Beyond, Lecture Notes in Physics, 9, World Scientific, 1987.

    MATH  Google Scholar 

  50. Odlyzko, A.M., D.J. Randall, On the Periods of Some Graph Transformations, Complex Systems, 1, 1987, 203–210.

    MathSciNet  MATH  Google Scholar 

  51. Pham Dinh Tao, S. El Bernoussi, Iterative Behaviour, Fixed Point of a Class of Monotone Operators. Application to Non-Symmetric Threshold Functions, Disc. Maths., 70, 1988, 85–101.

    Article  MATH  Google Scholar 

  52. Peliti, L. (ed), Disordered Systems and Biological Models, Procc of the Workshop and Disordered Systems and Biol. Modelling, Bogotá Colombia, 1987, World Scientific, CIF Series, 14, 1989.

    Google Scholar 

  53. Peretto, P., Collective Properties of Neural Networks: A Statistical Physics Approach, Biol. Cybern., 50, 1984, 51–62.

    Article  MATH  Google Scholar 

  54. Poljak, S., D. Turzik, On Pre-Periods of Discrete Influence Systems, Disc. Appi. Maths., 13, 1986, 33–39.

    Article  MathSciNet  MATH  Google Scholar 

  55. Poljak, S., D. Turzik, On an Application of Convexity to Discrete Systems, Disc. Appi. Math., 13, 1986, 27–32.

    Article  MathSciNet  MATH  Google Scholar 

  56. Rumelhart, D.E., J.L. McClelland (eds), Parallel and Distributed Processing: Explorations in the Micro structure of Cognition, MIT Press, 1986.

    Google Scholar 

  57. Robert, F., Discrete Iterations. A Metric Study, Springer Series in Computational Mathematics, Springer-Verlag, 1986.

    Book  MATH  Google Scholar 

  58. Robert, Y., M. Tchuente, Connection-Graph and Iteration-Graph of Monotone Boolean Functions, Disc. Appi. Maths., 11,1985, 245–253.

    Article  MathSciNet  Google Scholar 

  59. Shingai, R., Maximum Period of 2-Dimensional Uniform Neural Networks, Inf. and Control, 41, 1979, 324–341.

    Article  MathSciNet  MATH  Google Scholar 

  60. Shingai, R., The Maximum Period Realized in 1-D Uniform Neural Networks, Trans. IECE, Japan, E61, 1978, 804–808.

    Google Scholar 

  61. Smith, A.R., Simple Computation-Universal Cellular Spaces, J. ACM, 18(3), 1971, 339–353.

    Article  MATH  Google Scholar 

  62. Tchuente, M., Contribution a l’Etude des Methodes de Calcul pour des Systmes de Type Coopratif, Thesis, IMAG, Grenoble, France, 1982.

    Google Scholar 

  63. Tchuente, M., Evolution de Certains Automates Cellulaires Uniformes Binaires A Seuil, Seminaire 265, IMAG, Grenoble, 1977.

    Google Scholar 

  64. Toffoli, T., M. Margolus, Cellular Automata Machines: A New Environment for Modeling, MIT Press, 1987.

    Google Scholar 

  65. Toom, A.L., Monotonie Binary Cellular Automata, Problemy Peredaci Informacii, 12(1), 1976, 48–54.

    MathSciNet  MATH  Google Scholar 

  66. Toom, A.L., L.G. Mityushin, Two Results Regarding Noncomputability for Univariate Cellular Automata, Problemy Peredaci Informacii, 12(2), 1976, 69–75.

    MathSciNet  MATH  Google Scholar 

  67. Ulam S., On Some Mathematical Problems Connected with Patterns of Growth of Figures in Essays on Cellular Automata, A.W. Burks (ed), Univ. of Illinois Press, 1970, 219–243.

    Google Scholar 

  68. Vichniac, G., Simulating Physics with Cellular Automata, Physica 10D, 1984, 96–116.

    MathSciNet  Google Scholar 

  69. Vichniac G., Cellular Automata Models of Disordered and Organization in Disordered Systems ans Biol. Org., E. Bienenstock et al (eds), NATO ASI Series F, 20, 1986, 3–19.

    Google Scholar 

  70. von Neumann, J., Theory of Self-Reproducing Automata, A. W. Burks (ed), Univ. of Illinois Press, 1966.

    Google Scholar 

  71. Von Neumann, J., The General and Logical Theory of Automata in Hixon Synposium Proc., 1948 in J.N. Neumann Collected Works, A.H. Taub (ed), Pergamon Press, V,288–328, 1963.

    Google Scholar 

  72. Wolfram, S., Theory and Applications of Cellular Automata, World Scientific, 1986.

    MATH  Google Scholar 

  73. Wolfram, S., Universality and Complexity in Cellular Automata, Physica 10D, 1984, 1–35.

    MathSciNet  Google Scholar 

  74. Wolfram, S., Twenty Problems in the Theory of Cellular Automata, Phys. Scripta T9, 1985, 170.

    Article  MathSciNet  Google Scholar 

  75. Waksman, A., A Model of Replication, J.A.C.M., 16(1), 1966, 178–188.

    Google Scholar 

  76. Winograd, T., A Simple Algorithm for Self-Reproduction, MIT, Project MAC, Artificial Intelligence, Memo 198, 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Kluwer Academic Publishers

About this chapter

Cite this chapter

Goles, E., Martínez, S. (1990). Introduction. In: Neural and Automata Networks. Mathematics and Its Applications, vol 58. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0529-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-0529-0_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6724-9

  • Online ISBN: 978-94-009-0529-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics