Rule 184

Rule 184 is a one-dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems:

The apparent contradiction between these descriptions is resolved by different ways of associating features of the automaton's state with particles.

The name of Rule 184 is a Wolfram code that defines the evolution of its states. The earliest research on Rule 184 is by Li (1987) and Krug & Spohn (1988). In particular, Krug and Spohn already describe all three types of particle system modeled by Rule 184.

A state of the Rule 184 automaton consists of a one-dimensional array of cells, each containing a binary value (0 or 1). In each step of its evolution, the Rule 184 automaton applies the following rule to each of the cells in the array, simultaneously for all cells, to determine the new state of the cell:

An entry in this table defines the new state of each cell as a function of the previous state and the previous values of the neighboring cells on either side. The name for this rule, Rule 184, is the Wolfram code describing the state table above: the bottom row of the table, 10111000, when viewed as a binary number, is equal to the decimal number 184.

The rule set for Rule 184 may also be described intuitively, in several different ways:

From the descriptions of the rules above, two important properties of its dynamics may immediately be seen. First, in Rule 184, for any finite set of cells with periodic boundary conditions, the number of 1s and the number of 0s in a pattern remains invariant throughout the pattern's evolution. Rule 184 and its reflection are the only elementary cellular automata to have this property of number conservation. Similarly, if the density of 1s is well-defined for an infinite array of cells, it remains invariant as the automaton carries out its steps. And second, although Rule 184 is not symmetric under left-right reversal, it does have a different symmetry: reversing left and right and at the same time swapping the roles of the 0 and 1 symbols produces a cellular automaton with the same update rule.

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