Goles, E., & Moreira, A. (2012). NumberConserving Cellular Automata and Communication Complexity: A Numerical Exploration Beyond Elementary CAs. J. Cell. Autom., 7(2), 151–165.
Abstract: We perform a numerical exploration of numberconserving cellular automata (NCCA) beyond the class of elementary CAs, in search of examples with high communication complexity. We consider some possible generalizations of the elementary rule 184 (a minimal model of traffic, which is the only nontrivial elementary NCCA). as well as the classes of NCCAs which minimally extend either the radius or the state set (with respect to the 2 states and radius 1 of the elementary case). Both for 3 states and radius 1, and for 2 stales and radius 2, NCCA appear that are conjectured to have maximal (exponential) communication complexity. Examples are given also for (conjectured) linear and quadratic behaviour.

Goles, E., Guillon, P., & Rapaport, I. (2011). Traced communication complexity of cellular automata. Theor. Comput. Sci., 412(30), 3906–3916.
Abstract: We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansivity, entropy and give the asymptotic communication complexity of most elementary cellular automata. (C) 2011 Elsevier B.V. All rights reserved.

Goles, E., Meunier, P. E., Rapaport, I., & Theyssier, G. (2011). Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci., 412(12), 2–21.
Abstract: The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most cases. In this article, we introduce necessary conditions for a cellular automaton to be “universal”, according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of spacetime rescaling and it is intrinsic to the model of cellular automata. intrinsic universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed us to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, were not intrinsically universal. (C) 2010 Elsevier B.V. All rights reserved.

Goles, E., Moreira, A., & Rapaport, I. (2011). Communication complexity in numberconserving and monotone cellular automata. Theor. Comput. Sci., 412(29), 3616–3628.
Abstract: One third of the elementary cellular automata (CAs) are either numberconserving (NCCAs) or monotone (increasing or decreasing). In this paper we prove that, for all of them, we can find linear or constant communication protocols for the prediction problem. In other words, we are able to give a succinct description for their dynamics. This is not necessarily true for general NCCAs. In fact, we also show how to explicitly construct, from any CA, a new NCCA which preserves the original communication complexity. (C) 2011 Elsevier B.V. All rights reserved.
