In
mathematics and computation theory there are a class of cellular
automatons which are known as elementary automatons. This class of
cellular automatons is restricted to the one dimensional grid (in the
figures below the second dimension, ordinate (vertical) axis, is time)
with cells either on or off. Another important simplification is that
the actual state of the cell at given time, \( x_{i,t} \),
depends only on the previous state of the same cell and the previous
states of its immediate neighbors, i.e., on \( \{x_{i-1,t-1},x_{i,t-1},x_{i+1,t-1}\} \). Due to these restrictions
and simplifications, generally speaking cellular automatons might evolve
in the infinite dimensions, have infinite neighborhoods and have
limitless number of possible cell states, these cellular automatons
appear to be very simple, though as we show below they can replicate
very complex and even chaotic behavior.