# Cellular Automata/Examples on Rule 110
## Formal definition

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 $r=110$
* $B=\{b_{-1},b_{1}\}$ is the optional boundary usually $B=\{0,0\}$ chosen not to interfere with the quiescent background of all zeros

## De Bruijn diagrams
### Overlap

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

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

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

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

#### Quiescent background

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.
    $b_{L}=b_{R}=[1,0,0,1]$

#### Ether background

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.

## Listing preimages
### Bounded lattice

An example how to list preimages of the ether sequence on an bounded lattice
    $b_{L}=b_{R}=b_{u}$
Weights for the preimage network

### Garden of Eden sequences
#### 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
### the ether

    (00010011011111)*

## See also

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

## Reference notes

* ↑ Wolfram, Stephen (May 14, 2002). A New Kind of Science. online. Champaign, IL: Wolfram Media, Inc. p. 1168. ISBN 1-57955-008-8. OCLC 47831356. {{cite book}}: External link in |others= (help)