Efficient Computation in Brownian Cellular Automata

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


A Brownian cellular automaton is a kind of asynchronous cellular automaton, in which certain local configurations—like signals—propagate randomly in the cellular space, resembling Brownian motion. The Brownian-like behavior is driven by three kinds of local transition rules, two of which are locally reversible and rotation symmetric, thus mapping a rule’s left-hand side into a right-hand side is equivalent modulus rotations of multiples of 90 degrees. As a result, any update of cells using these rules can always be followed by a reversed update undoing it; this resembles the reversal of chemical reactions or other molecular processes. The third transition rule is not reversible and is merely used for diffusive purposes, so that signals can fluctuate forward and backward on wires like with random walks of molecules. The use of only these three rules is sufficient for embedding arbitrary asynchronous circuits on the cellular automaton, thus making it computationally universal. Key to this universality is the straightforward implementation of signal propagation as well as of the active backtracking of cell updates, which enables an effective realization of arbitration and choice – a functionality that is essential for asynchronous circuits but usually hard to implement efficiently on non-Brownian cellular automata. We show how to speed up the operation of circuits embedded in our Brownian cellular automaton. One method focuses on the design scheme of the circuits, by confining all necessary Brownian motions to local configurations representing primitive elements of circuits, such that a wire connecting two elements no longer needs backward propagation of signals on it, thus allowing the use of conventional design schemes of asynchronous circuits without change. Another method is to implement ratchets on the input and output lines by using various configurations in the cellular space, so as to further increase the speeds of signals on the wires.


Cellular Automaton Transition Rule Primitive Element Output Line Cellular Automaton Model 
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  1. 1.
    Adachi, S., Peper, F., Lee, J.: Computation by asynchronously updating cellular automata. Journal of Statistical Physics 114(1/2), 261–289 (2004)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bennett, C.H.: The thermodynamics of computation–a review. International Journal of Theoretical Physics 21(12), 905–940 (1982)CrossRefGoogle Scholar
  3. 3.
    Dasmahapatra, S., Werner, J., Zauner, K.P.: Noise as a computational resource. Int. J. of Unconventional Computing 2(4), 305–319 (2006)Google Scholar
  4. 4.
    Frank, M., Vieri, C., Ammer, M.J., Love, N., Margolus, N.H., Knight Jr., T.: A scalable reversible computer in silicon. In: Calude, C.S., Casti, J., Dinneen, M.J. (eds.) Unconventional Models of Computation, pp. 183–200. Springer, Singapore (1998)Google Scholar
  5. 5.
    Hauck, S.: Asynchronous design methodologies: an overview. Proc. IEEE 83(1), 69–93 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Heinrich, A.J., Lutz, C.P., Gupta, J.A., Eigler, D.M.: Molecule cascades. Science 298, 1381–1387 (2002)CrossRefGoogle Scholar
  7. 7.
    Kish, L.B.: Thermal noise driven computing. Applied Physics Letters 89(14), 144104–1–3 (2006)Google Scholar
  8. 8.
    Lee, J., Peper, F., Adachi, S., Morita, K., Mashiko, S.: Reversible computation in asynchronous cellular automata. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002. LNCS, vol. 2509, pp. 220–229. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Lee, J., Adachi, S., Peper, F., Morita, K.: Embedding universal delay-insensitive circuits in asynchronous cellular spaces. Fundamenta Informaticae 58(3-4), 295–320 (2003)MathSciNetMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    Lee, J., Peper, F.: On Brownian cellular automata. In: Theory and Applications of Cellular Automata, p. 278. Luniver Press (2008)Google Scholar
  13. 13.
    Lee, J., Peper, F., et al.: Brownian circuits—part II (in preparation)Google Scholar
  14. 14.
    MacLennan, B.J.: Computation and nanotechnology. Int. J. of Nanotechnology and Molecular Computation 1 (2009)Google Scholar
  15. 15.
    Patra, P., Fussell, D.S.: Conservative delay-insensitive circuits. In: Workshop on Physics and Computation, pp. 248–259 (1996)Google Scholar
  16. 16.
    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
  17. 17.
    Peper, F., Lee, J., Isokawa, T.: Cellular nanocomputers: a focused review. Int. J. of Nanotechnology and Molecular Computation 1, 33–49 (2009)Google Scholar
  18. 18.
    Peper, F., Lee, J., et al.: Brownian circuits—Part I (in preparation)Google Scholar
  19. 19.
    Yanagida, T., Ueda, M., Murata, T., Esaki, S., Ishii, Y.: Brownian motion, fluctuation and life. Biosystems 88(3), 228–242 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Tokyo 2010

Authors and Affiliations

  • Jia Lee
    • 1
    • 2
  • Ferdinand Peper
    • 2
  1. 1.College of Computer ScienceChongqing UniversityChongqingChina
  2. 2.National Institute of Information and Communications Technology (NiCT)Nano ICT GroupKobeJapan

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