Cellular Automata/Examples on Rule 110

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Formal definition[edit]

The commonly known 1D binary CA rule 110 is defined as  \langle Z, S, N, f, B \rangle where

  • Z can be finite or infinite
  • S=\{ 0,1 \} is a set of two values
  • N=\{ -1,0,1 \} is the neighborhood of size k=3 with symmetric radius k_0=1
  • f is the local transition function rule r=110
000 -> 0
001 -> 1
010 -> 1
011 -> 1
100 -> 0
101 -> 1
110 -> 1
111 -> 0
  • B=\{ b_{-1},b_{1} \} is the optional boundary usually B=\{ 0,0 \} chosen not to interfere with the quiescent background of all zeros

De Bruijn diagrams[edit]

Overlap[edit]

Neighborhoods of adjacent cells overlapping

Neighborhoods of adjacent cells are overlapping for k-1=2 cells. There are |S|^{k-1}=4 different overlaps {00,01,10,11} or written in compact form {0,1,2,3}.

Symbolic De Bruijn diagram[edit]

Symbolic De Bruijn diagram for rule 110

The De Bruijn diagram has |S|^{k-1}=4 nodes (one for each of the possible overlaps) and |S|^k=8 links (one for each of the possible neighborhoods).

 D=
\begin{bmatrix}
0 & 1 & \cdot & \cdot \\
\cdot & \cdot & 1 & 1 \\
0 & 1 & \cdot & \cdot \\
\cdot & \cdot & 1 & 0
\end{bmatrix}

preimage matrix[edit]

There are two preimage matrices, one for each of the available cell states.

 D(0)=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}

\qquad D(1)=
\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}

Boundary conditions[edit]

There are two commonly used backgrounds, the quiescent background and the ether.

Quiescent background[edit]

The quiescent background is an infinite sequence with period \alpha=\overline{0} of length |\alpha|=1 cell.

There are always exactly two preimages for this background independently on the length of the sequence from a single cell to infinity (see the preimage network). The left and right boundary vectors are equal.

 b_L = b_R = [1,0,0,1]

Preimage network for rule 110 and the quiescent background configuration

Ether background[edit]

The ether background is an infinite sequence with period \alpha=\overline{00010011011111} of length |\alpha|=14 cells. This is the prevailing background emerging from a random initial configuration.

The number of preimages of the ether configuration increases exponentially with the sequence length l going to infinity. A circular lattice is used to calculate the number of preimages of the period \alpha.

 p = p_c(\alpha)^{l/|\alpha|} = 2^{l/14}

Because of the exponential growth the boundary vector does not represent the number of preimages of the whole infinite background but only weights derived from the period's preimages. The value of the boundary vector depends on the position inside the period, in the next table vectors are columns for each of the 14 positions.

 overlaps | boundary vectors
---------------------------------------------------------------------
 00       | 0   0   0   0   2   2   0   0   1   0   0   0   0   0
 01       | 0   0   0   0   0   0   2   0   0   1   1   0   2   0
 10       | 0   0   0   2   0   0   0   1   0   1   0   2   0   0
 11       | 2   2   2   0   0   0   0   1   1   0   1   0   0   2
---------------------------------------------------------------------
 sequence |   0   0   0   1   0   0   1   1   0   1   1   1   1   1

Preimage network for rule 110 and the ether configuration

Listing preimages[edit]

Bounded lattice[edit]

An example how to list preimages of the ether sequence on an bounded lattice

b_L=b_R=b_u
 overlaps | backward preimage count vectors
----------------------------------------------------------------------
 00       | 0   0   0   0   7   7   7   0   4   4   3   2   2   1   1
 01       | 0   0   0   7   0   0   4   7   0   5   4   3   2   2   1
 10       | 0   0   0   0   7   7   7   0   4   4   3   2   2   1   1
 11       | 7   7   7   7   0   0   0   4   3   3   2   2   1   1   1
----------------------------------------------------------------------
 sequence |   0   0   0   1   0   0   1   1   0   1   1   1   1   1
 overlaps | forward preimage count vectors
----------------------------------------------------------------------
 00       | 1   2   2   2   0   1   1   0   0   1   0   0   0   0   0
 01       | 1   0   0   0   2   0   0   1   0   0   1   1   1   2   2
 10       | 1   0   0   0   1   0   0   0   1   0   1   1   2   2   3
 11       | 1   1   1   1   0   0   0   0   1   1   0   1   1   1   2
----------------------------------------------------------------------
 sequence |   0   0   0   1   0   0   1   1   0   1   1   1   1   1

Weigths for the preimage network

 overlaps | neighborhood (link) weigths
----------------------------------------
 000      | 0 0 0 0 0 7 0 0 0 0 0 0 0 0
 001      | 0 0 0 0 0 0 7 0 0 4 0 0 0 0
 010      | 0 0 0 0 0 0 0 4 0 0 2 2 1 2
 011      | 0 0 0 0 0 0 0 3 0 0 2 1 1 2
 100      | 0 0 0 0 7 0 0 0 4 0 0 0 0 0
 101      | 0 0 0 0 0 0 0 0 0 0 3 2 4 2
 110      | 0 0 0 7 0 0 0 0 0 3 0 2 1 1
 111      | 7 7 7 0 0 0 0 0 3 0 0 0 0 0
----------------------------------------
 sequence | 0 0 0 1 0 0 1 1 0 1 1 1 1 1
 overlaps | boundary vectors
----------------------------------------------------------------------
 00       | 0   0   0   0   0   7   7   0   0   4   0   0   0   0   0
 01       | 0   0   0   0   0   0   0   7   0   0   4   3   2   4   2
 10       | 0   0   0   0   7   0   0   0   4   0   3   2   4   2   3
 11       | 7   7   7   7   0   0   0   0   3   3   0   2   1   1   2
----------------------------------------------------------------------
 sequence |   0   0   0   1   0   0   1   1   0   1   1   1   1   1

Preimage network for rule 110 and the ether sequence on unrestricted boundaries

cyclic lattice[edit]

 overlaps | preimage count matrices
---------------------------------------------------------------------------------------
 00       | 0000 0000 0000 0000 0232 0232 0232 0000 0121 0121 0111 0110 0011 0100 1000
 01       | 0000 0000 0000 0232 0000 0000 0121 0232 0000 0221 0121 0111 0110 0011 0100
 10       | 0000 0000 0000 0000 0232 0232 0232 0000 0121 0121 0111 0110 0011 0100 0010
 11       | 0232 0232 0232 0232 0000 0000 0000 0121 0111 0111 0110 0011 0100 0010 0001
---------------------------------------------------------------------------------------
 sequence |     0    0    0    1    0    0    1    1    0    1    1    1    1    1
 overlaps | boundary vectors
---------------------------------------------------------------------
 00       | 0   0   0   0   2   2   0   0   1   0   0   0   0   0
 01       | 0   0   0   0   0   0   2   0   0   1   1   0   2   0
 10       | 0   0   0   2   0   0   0   1   0   1   0   2   0   0
 11       | 2   2   2   0   0   0   0   1   1   0   1   0   0   2
---------------------------------------------------------------------
 sequence |   0   0   0   1   0   0   1   1   0   1   1   1   1   1

Preimage network for rule 110 and the ether configuration on cyclyc boundaries

Subset diagram transition table[edit]

from |  to he left         |  to the right
--------------------------------------------------
0000 |  <-0-0000 <-1-0000  |  0000-0-> 0000-1->
0001 |  <-0-0001 <-1-0100  |  0001-0-> 0010-1->
0010 |  <-0-0000 <-1-0101  |  1000-0-> 0100-1->
0011 |  <-0-0001 <-1-0201  |  1001-0-> 0110-1->
0100 |  <-0-0000 <-1-1010  |  0000-0-> 0011-1->
0101 |  <-0-0001 <-1-1110  |  0001-0-> 0021-1->
0110 |  <-0-0000 <-1-1111  |  1000-0-> 0111-1->
0111 |  <-0-0001 <-1-1211  |  1001-0-> 0121-1->
1000 |  <-0-1010 <-1-0000  |  1000-0-> 0100-1->
1001 |  <-0-1011 <-1-0100  |  1001-0-> 0110-1->
1010 |  <-0-1010 <-1-0101  |  2000-0-> 0200-1->
1011 |  <-0-1011 <-1-0201  |  2001-0-> 0210-1->
1100 |  <-0-1010 <-1-1010  |  1000-0-> 0111-1->
1101 |  <-0-1011 <-1-1110  |  1001-0-> 0121-1->
1110 |  <-0-1010 <-1-1111  |  2000-0-> 0211-1->
1111 |  <-0-1011 <-1-1211  |  2001-0-> 0221-1->

Garden of Eden sequences[edit]

regular expression (the expression is left right symmetric)
0*1(00*1+1(1+00)*010)*1(1+00)*011(0+1)*
the shortest GoE sequence
01010

Gliders[edit]

the ether
(00010011011111)*

See also[edit]

  1. Rule 110 at MathWorld
  2. Rule 110 at Wolfram Atlas
  3. Harold V. Mcintosh, Rule 110 as it rules relates to the presence of gliders
  4. Rule 110 at Wikipedia