The Rule 110 cellular automaton (often called 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 with a particular repeating background pattern is known to be Turing complete. This implies that, in principle, any calculation or computer program can be simulated using this automaton.
Definition
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. This is the Wolfram code naming scheme.
History
In 2004, Matthew Cook published a proof that Rule 110 with a particular repeating background pattern 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 publication 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 several years.
Interesting properties
Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been directly proven, although proofs for several similar rules follow as simple corollaries (e.g. 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.
Matthew Cook proved Rule 110 capable of supporting universal computation by successively emulating cyclic tag systems, then 2-tag system, and then Turing machines. The final stage has exponential time overhead because the Turing machine's tape is encoded with a unary numeral system. Neary and Woods (2006) presented a different construction that replaces 2-tag systems with clockwise Turing machines and has polynomial overhead.
The proof of universality
Matthew Cook presented his proof of the universality of Rule 110 at a Santa Fe Institute conference, held before the publication of A New Kind of Science. Wolfram Research claimed that this presentation violated Cook's nondisclosure agreement with his employer, and obtained a court order excluding Cook's paper from the published conference proceedings. The existence of Cook's proof nevertheless became known. Interest in his proof stemmed not so much from its result as from its methods, specifically from the technical details of its construction. The character of Cook's proof differs considerably from the discussion of Rule 110 in A New Kind of Science. Cook has since written a paper setting out his complete proof.
Cook proved that Rule 110 was universal (or Turing complete) by showing it was possible to use the rule to emulate another computational model, the cyclic tag system, which is known to be universal. He first isolated a number of spaceships, self-perpetuating localized patterns, that could be constructed on an infinitely repeating pattern in a Rule 110 universe. He then devised a way for combinations of these structures to interact in a manner that could be exploited for computation.
Spaceships in Rule 110
The function of the universal machine in Rule 110 requires a finite number of localized patterns to be embedded within an infinitely repeating background pattern. The background pattern is fourteen cells wide and repeats itself exactly every seven iterations. The pattern is 00010011011111.
Three localized patterns are of particular importance in the Rule 110 universal machine. They are shown in the image below, surrounded by the repeating background pattern. The leftmost structure shifts to the right two cells and repeats every three generations. It comprises the sequence 0001110111 surrounded by the background pattern given above, as well as two different evolutions of this sequence.
In the figures, time elapses from top to bottom: the top line represents the initial state, and each following line the state at the next time.