(Internal) transformations on the space Σ of automaton configurations are defined as bi-infinite sequences of permutations of the cell symbols. A pair of transformations (γ,θ) is said to be an internal symmetry of a cellular automaton f:Σ→Σ if f=θ−1fγ⁠. It is shown that the full group of internal symmetries of an automaton f can be encoded as a group homomorphism F such that θ=F(γ)⁠. The domain and image of the homomorphism F have, in general, infinite order and F is presented by a local automaton-like rule. Algorithms to compute the symmetry homomorphism F and to classify automata by their symmetries are presented. Examples on the types of dynamical implications of internal symmetries are discussed in detail.

Topics
Theoretical computer science, Group theory
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© 1997 American Institute of Physics.
1997
American Institute of Physics
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