Communication Complexity and Intrinsic Universality in Cellular Automata
Résumé
Let F be a cellular automaton (CA). This paper establishes necessary conditions for F in order to be intrinsically universal. The central idea is to consider the communication complexity of various ``canonical problems'' related to the dynamics of F. We show that the intrinsic universality of F implies high communication complexity for each of the canonical problems. This result allows us to rule out many CAs from being intrinsically universal: The linear CAs, the expansive CAs, the reversible CAs and the elementary CAs 218, 33 and 94. Our communicational approach provides a finer tool than the one given by classical computational complexity analysis. In fact, we prove that for two of the canonical problems there exists a CA for which the computational complexity is maximal (P-complete, or Pi_1-complete) while the corresponding communication complexity is rather low. We also show the orthogonality of the problems. More precisely, for any pair of problems there exists a CA with low communication complexity for one but high communication complexity for the other.
Domaines
Mathématique discrète [cs.DM]
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