Von Neumann cellular automaton

Von Neumann cellular automata are the original expression of cellular automata, the development of which were prompted by suggestions made to John von Neumann by his close friend and fellow mathematician Stanislaw Ulam. Their original purpose was to provide insight into the logical requirements for machine self-replication and were used in von Neumann's universal constructor.

Nobili's cellular automaton is a variation of von Neumann's cellular automaton, augmented with the ability for confluent cells to cross signals and store information. The former requires an extra three states, hence Nobili's cellular automaton has 32 states, rather than 29. Hutton's cellular automaton is yet another variation, which allows a loop of data, analogous to Langton's loops, to replicate.

In general, cellular automata (CA) constitute an arrangement of finite state automata (FSA) that sit in positional relationships between one another, each FSA exchanging information with those other FSAs to which it is positionally adjacent. In von Neumann's cellular automaton, the finite state machines (or cells) are arranged in a two-dimensional Cartesian grid, and interface with the surrounding four cells. As von Neumann's cellular automaton was the first example to use this arrangement, it is known as the von Neumann neighbourhood.

The set of FSAs define a cell space of infinite size. All FSAs are identical in terms of state-transition function, or rule-set.

The neighborhood (a grouping function) is part of the state-transition function, and defines for any cell the set of other cells upon which the state of that cell depends.

All cells transition state synchronously, in step with a universal "clock" as in a synchronous digital circuit.

Each FSA of the von Neumann cell space can accept any of the 29 states of the rule-set. The rule-set is grouped into five orthogonal subsets. Each state includes the colour of the cell in the cellular automata program Golly in (red, green, blue). They are

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