A universal map is derived for all deterministic 1D cellular automata (CA)
containing no freely adjustable parameters. The map can be extended to an
arbitrary number of dimensions and topologies and its invariances allow to
classify all CA rules into equivalence classes. Complexity in 1D systems is
then shown to emerge from the weak symmetry breaking of the addition modulo an
integer number p. The latter symmetry is possessed by certain rules that
produce Pascal simplices in their time evolution. These results elucidate
Wolfram's classification of CA dynamics.
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