Cellular Automata/Examples on Rule 110

(Redirected from Cellular Automata/Rule 110)

Formal definition

edit

The commonly known 1D binary CA rule 110 is defined as   where

  •   can be finite or infinite
  •   is a set of two values
  •   is the neighborhood of size   with symmetric radius  
  •   is the local transition function rule  
000 -> 0
001 -> 1
010 -> 1
011 -> 1
100 -> 0
101 -> 1
110 -> 1
111 -> 0
  •   is the optional boundary usually   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   cells. There are   different overlaps   or written in compact form  .

Symbolic De Bruijn diagram

edit
 
Symbolic De Bruijn diagram for rule 110

The De Bruijn diagram has   nodes (one for each of the possible overlaps) and   links (one for each of the possible neighborhoods).

 

preimage matrix

edit

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

 

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   of length   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.

 

 

Ether background

edit

The ether background is an infinite sequence with period   of length   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   going to infinity. A circular lattice is used to calculate the number of preimages of the period  .

 

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

 

Listing preimages

edit

Bounded lattice

edit

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

 
 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

Weights for the preimage network

 overlaps | neighborhood (link) weights
----------------------------------------
 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

 

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

 

Subset diagram transition table

edit
from |  to the 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)*[citation needed]
the shortest GoE sequence
01010[1]

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

Reference notes

edit
  1. 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)