Advertisement

On Some Dynamical Properties of Monotone Networks

  • Yves Robert
  • Maurice Tchuente
Part of the NATO ASI Series book series (volume 20)

Abstract

Cellular automata were first introduced in the 1950’s by John Von Neumann. Informally, a cellular automaton can be viewed as a discrete dynamical system whose global behaviour is generated by the (simple) local interactions of its elementary cells, hereafter called the sites of the automaton. As pointed out by Wolfram (1984), cellular automata have arisen in several disciplines, because they provide examples in which the generation of complex behaviour by the cooperative effects of simple components may be studied. No unified mathematical framework has been yet developped to modelize the iterative behaviour of general networks of automata, but some tools such as algebraic operators and Lyapounov functions (Goles (1985), Fogelman (1985)), modular arithmetic and polynomial algebra (Martin (1983)), arithmetic in finite fields (Gill (1966), Tchuente (1982)) have been introduced to analyze special classes of cellular automata. In this paper, we characterize the dynamics of some monotone cellular automata. First we use a morphism technique to derive the iterative behaviour of automata whose transition functions are generalized majority functions. Then, we study some relationships between the connection-graph and the iteration-graph of boolean automata with memory, assuming that the transition function of each site is monotone with regard to its inputs.

Keywords

Transition Function Cellular Automaton Boolean Network Elementary Circuit Polynomial Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. FOGELMAN F, Thesis, Grenoble (1985)Google Scholar
  2. GOLES E., Thesis, Grenoble (1985)Google Scholar
  3. GILL A., Linear sequential circuits, Analysis, Synthesis and Applications, Mc Graw-Hill Series in Computer Science, New York, 1966MATHGoogle Scholar
  4. MARTIN O., Odlyzko A., Wolfram S., Algebraic properties of cellular automata, Bell Laboratories Report (1983)Google Scholar
  5. ROBERT Y., TCHUENTE M., Connection-graph and iteration-graph of monotone boolean functions, Discrete Applied Mathematics 11 (1985)Google Scholar
  6. ROBERT Y., TCHUENTE M., Dynamical Behaviour of monotone networks, RR 521, IMAG Grenoble, March 1985Google Scholar
  7. TCHUENTE M., Thesis, Grenoble (1982)Google Scholar
  8. WOLFRAM S., Preface to Physica 10D (1984), Elsevier Science PublishersGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Yves Robert
    • 1
  • Maurice Tchuente
    • 1
  1. 1.CNRS — Laboratoire TIM3/IMAGSt Martin d’Hères CedexFrance

Personalised recommendations