 
Summary: Notes on cellular automata
J.P. Allouche, M. Courbage, G. Skordev
1 Introduction
Cellular automata were introduced by J. von Neumann (see 29]) after a suggestion of S. Ulam
147, p. 274]. They are a selfreproducing model, that was designed in order to answer the question
\is it possible to construct robots that can construct identical robots, i.e., robots with the same
\complexity"?". The model proposed by von Neumann gives a positive answer to this question.
Another \philosophical" background is the production of order from chaos and the concept of
\selforganization" (see for example 11], see also 125]).
It is of course tempting to see life itself behind selfreproduction. This might be the reason for
the choice of many expressions in this theory: cells, living or dead structures, garden of Eden, game
of Life...
2 The game of Life
The most popular example of cellular automaton is the socalled \Game of Life" introduced by
Conway in the 70's. The reference given for example in 156, p. 66] is: J. H. Conway, 1970 unpub
lished. Other references are the articles of M. Gardner in 1970{1972 in Scienti c American, see for
example 68], and the books 19, ch. 25] and 30]. Note that this game is named after Conway and
Golay in 120] (Golay's game has an underlying hexagonal tiling according to 69]).
This game is de ned as follows. We have an in nite twodimensional board whose elementary
squares are called \cells". A cell can be \living" or \dead". The neighbors of a cell are de ned to
