Asynchronous cellular automaton

Cellular automata, as with other multi-agent system models, usually treat time as discrete and state updates as occurring synchronously. The state of every cell in the model is updated together, before any of the new states influence other cells. In contrast, an asynchronous cellular automaton is able to update individual cells independently, in such a way that the new state of a cell affects the calculation of states in neighbouring cells.

Implementations of synchronous updating can be analysed in two phases. The first, interaction, calculates the new state of each cell based on the neighbourhood and the update rule. State values are held in a temporary store. The second phase updates state values by copying the new states to the cells.

In contrast, asynchronous updating does not necessarily separate these two phases: in the simplest case (fully asynchronous updating), changes in state are implemented immediately.

The synchronous approach assumes the presence of a global clock to ensure all cells are updated together. While convenient for preparing computer systems, this might be an unrealistic assumption if the model is intended to represent, for example, a living system where there is no evidence of the presence of such a device.

A general method repeatedly discovered independently (by K. Nakamura in the 1970s, by T. Toffoli in the 1980s, and by C. L. Nehaniv in 1998) allows one to emulate exactly the behaviour of a synchronous cellular automaton via an asynchronous one constructed as a simple modification of the synchronous cellular automaton (Nehaniv 2002). Correctness of this method however has only more recently been rigorously proved (Nehaniv, 2004). As a consequence, it follows immediately from results on synchronous cellular automata that asynchronous cellular automata are capable of emulating, e.g., Conway's Game of Life, of universal computation, and of self-replication (e.g., as in a Von Neumann universal constructor). Moreover, the general construction and the proof also applies to the more general class of synchronous automata networks (inhomogeneous networks of automata over directed graphs, allowing external inputs – which includes cellular automata as a special case), showing constructively how their behaviour may be asynchronously realized by a corresponding asynchronous automata network.

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