Universal 2-State Asynchronous Cellular Automaton with Inner-Independent Transitions

  • Susumu Adachi
  • Jia Lee
  • Ferdinand Peper
Conference paper
Part of the Proceedings in Information and Communications Technology book series (PICT, volume 2)


This paper proposes a computationally universal square lattice asynchronous cellular automaton, in which cells have merely two states. The transition function according to which a cell is updated takes as its arguments the states of the cells at orthogonal or diagonal distances 1 or 2 from the cell. The proposed cellular automaton is inner-independent—a property according to which a cell’s state does not depend on its previous state, but merely on the states of cells in its neighborhood. Playing a role in classical spin systems, inner-dependence has only been investigated in the context of synchronous cellular automata. The asynchronous update mode used in this paper allows an update of a cell state to take place—but only so with a certain probability—whenever the cell’s neighborhood states matches an element of the transition function’s domain. Universality of the model is proven through the construction of three circuit primitives on the cell space, which are universal for the class of Delay-Insensitive circuits.


Cellular Automaton Cellular Automaton Transition Rule Cell Space Cellular Automaton Model 
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  1. 1.
    Adachi, S., Peper, F., Lee, J.: Computation by asynchronously updating cellular automata. J. Stat. Phys. 114(1/2), 261–289 (2004)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Adachi, S., Peper, F., Lee, J.: Universality of Hexagonal Asynchronous Totalistic Cellular Automata. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds.) ACRI 2004. LNCS, vol. 3305, pp. 91–100. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Adachi, S., Lee, J., Peper, F.: On signals in asynchronous cellular spaces. IEICE Trans. inf. & syst. E87-D(3), 657–668 (2004)MathSciNetGoogle Scholar
  4. 4.
    Adachi, S., Lee, J., Peper, F., Umeo, H.: Kaleidoscope of Life: a 24-neighborhood outer-totalistic cellular automaton. Physica D 237, 800–817 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Wining Ways For Your Mathematical Plays, vol. 2. Academic Press, New York (1982)Google Scholar
  6. 6.
    Hauck, S.: Asynchronous design methodologies: an overview. Proc. IEEE 83(1), 69–93 (1995)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ilachinski, A.: Cellular Automata. World Scientific Publishing, Singapore (2001)MATHGoogle Scholar
  8. 8.
    Ingerson, T.E., Buvel, R.L.: Structures in asynchronous cellular automata. Physica D 10, 59–68 (1984)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Keller, R.M.: Towards a theory of universal speed-independent modules. IEEE Trans. Comput. C-23(1), 21–33 (1974)CrossRefGoogle Scholar
  10. 10.
    Lee, J., Adachi, S., Peper, F., Morita, K.: Embedding universal delay-insensitive circuits in asynchronous cellular spaces. Fund. Inform. 58(3/4), 295–320 (2003)MathSciNetMATHGoogle Scholar
  11. 11.
    Lee, J., Adachi, S., Peper, F., Mashiko, S.: Delay-insensitive computation in asynchronous cellular automata. Journal of Computer and System Sciences 70, 201–220 (2005)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Lee, J., Peper, F., Adachi, S., Mashiko, S.: Universal Delay-Insensitive Systems With Buffering Lines. IEEE Trans. Circuits and Systems 52(4), 742–754 (2005)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lee, J., Peper, F., Adachi, S., Morita, K.: An Asynchronous Cellular Automaton Implementing 2-State 2-Input 2-Output Reversed-Twin Reversible Elements. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) ACRI 2008. LNCS, vol. 5191, pp. 67–76. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Lee, J., Peper, F.: On brownian cellular automata. In: Proc. of Automata 2008, UK, pp. 278–291. Luniver Press (2008)Google Scholar
  15. 15.
    von Neumann, J.: The Theory of Self-Reproducing Automata, edited and completed by A. W. Burks. University of Illinois Press, Urbana (1966)Google Scholar
  16. 16.
    Patra, P., Fussell, D.S.: Efficient building blocks for delay insensitive circuits. In: Proceedings of the International Symposium on Advanced Research in Asynchronous Circuits and Systems, pp. 196–205. IEEE Computer Society Press, Silver Spring (1994)CrossRefGoogle Scholar
  17. 17.
    Peper, F., Lee, J., Adachi, S., Mashiko, S.: Laying out circuits on asynchronous cellular arrays: a step towards feasible nanocomputers? Nanotechnology 14(4), 469–485 (2003)CrossRefGoogle Scholar
  18. 18.
    Peper, F., Lee, J., Abo, F., Isokawa, T., Adachi, S., Matsui, N., Mashiko, S.: Fault-Tolerance in Nanocomputers: A Cellular Array Approach. IEEE Trans. Nanotech. 3(1), 187–201 (2004)CrossRefGoogle Scholar
  19. 19.
    Wolfram, S.: Cellular Automata and Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar

Copyright information

© Springer Tokyo 2010

Authors and Affiliations

  • Susumu Adachi
    • 1
  • Jia Lee
    • 1
    • 2
  • Ferdinand Peper
    • 1
  1. 1.National Institute of Information and Communications TechnologyNano ICT GroupJapan
  2. 2.College of Computer ScienceChong Qing UniversityChina

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