## LOCAL NESTED STRUCTURE IN RULE 30

Citations: | 2 - 1 self |

### BibTeX

@MISC{Rowland_localnested,

author = {Eric S. Rowland},

title = {LOCAL NESTED STRUCTURE IN RULE 30},

year = {}

}

### OpenURL

### Abstract

Abstract. At row 2 n in the cellular automaton rule 30, a region of the initial condition reappears on the right side, which causes the automaton to “begin again ” locally. As a result, local nested structure is produced. This phenomenon is ultimately due to the property that rule 30 is reversible in time under the condition that the right half of each row is white. The main result of the paper establishes the presence of local nested structure in k-color rules with this bijectivity property, and we explore a class of integer sequences characterizing the nested structure. We also prove an observation of Wolfram regarding the period length doubling of diagonals on the left side of rule 30. 1.

### Citations

220 |
Universality and complexity in cellular automata
- Wolfram
- 1984
(Show Context)
Citation Context ...it would appear that by examining {aR(n)} one can distinguish both reducible nested (O(log t)) behavior from irreducible uniformly disordered “class 3” (O(t 2 )) behavior (in Wolfram’s classification =-=[6, 5]-=-) and also reducible uniform (O(t log t)) behavior from irreducible uniform (O(t 2 )) behavior. We proceed to consider rules with other values of k and d. The 2-color, range [−1, 0] right bijective ru... |

214 |
Statistical mechanics of cellular automata
- Wolfram
- 1983
(Show Context)
Citation Context ...ion 5 we prove Wolfram’s observation on the period doubling of left diagonals by studying conditions under which rule 30 is right bijective. 2. Bijectivity and convergence We adopt Wolfram’s paradigm =-=[6, 4]-=- for studying cellular automata. In particular, we use his numbering convention for cellular automaton rules (in which the rule icon is read in base k) with the following additional notation. The set ... |

107 |
A New Kind of Science (Wolfram
- Wolfram
- 2002
(Show Context)
Citation Context ...e part of the initial condition. Figure 2 shows the beginning of rule 30—a triangular pattern of more or less uniform disorder. As convincingly established by Stephen Wolfram in A New Kind of Science =-=[6]-=-, systems like rule 30 defy the intuition that simple programs only produce simple results. Rule 30, despite its simple definition, does not appear to generate a regular pattern. However, there are ce... |

18 |
Aperiodicity in one-dimensional cellular automata, Phys
- JEN
- 1990
(Show Context)
Citation Context ...his numbering convention for cellular automaton rules (in which the rule icon is read in base k) with the following additional notation. The set of k colors will be denoted [k], with the special case =-=[2]-=- = {�, �}. The set [k] Z of doubly infinite sequences of cells (indexed by the set of integers Z, increasing to the right) is the set of rows. For a row R ∈ [k] Z , let R(m) be the color of the cell a... |

8 |
Global Properties of Cellular Automata
- Jen
- 1986
(Show Context)
Citation Context ... . . .,xd2) for every (d2 −d1+1)-tuple (xd1, xd1+1, . . .,xd2). We say that f is left bijective if its left–right reflection is right bijective. This positional bijectivity has been considered by Jen =-=[1, 2]-=- and by Wolfram [6] (under the name “one-sided additivity”). The results in this paper are stated for right bijective rules but hold for left bijective rules mutatis mutandis. Accordingly, we primaril... |

1 |
BijectiveRules [a package for Mathematica] (available from http://math.rutgers.edu/∼erowland/programs.html
- Rowland
(Show Context)
Citation Context ... right nestedness of rules 30 and 90 is a consequence of the left bijectivity of these rules. A Mathematica package for studying left and right bijective rules is available from the author’s web site =-=[3]-=-. The nested structure of rule 30 is more easily seen when each row is shifted right relative to the one below it. After an additional left–right reflection (to put it into the standard form used in l... |