Rule 110 cellular automaton

The Rule 110 cellular automaton (often simply Rule 110) is an elementary cellular automaton with interesting behavior on the boundary between stability and chaos. In this respect, it is similar to Conway's Game of Life. Like Life, Rule 110 is known to be Turing complete. This implies that, in principle, any calculation or computer program can be simulated using this automaton.

In an elementary cellular automaton, a one-dimensional pattern of 0s and 1s evolves according to a simple set of rules. Whether a point in the pattern will be 0 or 1 in the new generation depends on its current value, as well as on those of its two neighbors. The Rule 110 automaton has the following set of rules:

The name "Rule 110" derives from the fact that this rule can be summarized in the binary sequence 01101110; interpreted as a binary number, this corresponds to the decimal value 110.

Around 2000, Matthew Cook published a proof that Rule 110 is Turing complete, i.e., capable of universal computation, which Stephen Wolfram had conjectured in 1985. Cook presented his proof at the Santa Fe Institute conference CA98 before the publishing of Wolfram's book, A New Kind of Science. This resulted in a legal affair based on a non-disclosure agreement with Wolfram Research. Wolfram Research blocked publication of Cook's proof for 2 years.

Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been proven, although proofs for several similar rules should follow as simple corollaries. For instance, Rule 124, which is the horizontal reflection of Rule 110. Rule 110 is arguably the simplest known Turing complete system.

Rule 110, like the Game of Life, exhibits what Wolfram calls "Class 4 behavior", which is neither completely stable nor completely chaotic. Localized structures appear and interact in complex ways.

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