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.

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