Communication Complexity and Intrinsic Universality in Cellular Automata

Eric Goles Chacc (1) , Pierre-Etienne Meunier (2) , Ivan Rapaport (3) , Guillaume Theyssier (2)

1 Facultad de Ingeniería y Ciencias [Santiago]
2 LAMA - Laboratoire de Mathématiques
3 DIM - Departamento de Ingeniería Matemática [Santiago]

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]

Eric Goles Chacc, Pierre-Etienne Meunier, Ivan Rapaport, Guillaume Theyssier. Communication Complexity and Intrinsic Universality in Cellular Automata. 2009. ⟨hal-00440186v1⟩