Communication complexity and intrinsic universality in cellular automata

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 space-time 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.

[1]  B. Durand,et al.  The Game of Life: Universality Revisited , 1999 .

[2]  Matthew Cook,et al.  Universality in Elementary Cellular Automata , 2004, Complex Syst..

[3]  Mathieu Sablik.  Directional dynamics for cellular automata: A sensitivity to initial condition approach , 2008, Theor. Comput. Sci..

[4]  Guillaume Theyssier,et al.  ON LOCAL SYMMETRIES AND UNIVERSALITY IN CELLULAR , 2009 .

[5]  Eric Goles Ch.,et al.  Understanding a Non-trivial Cellular Automaton by Finding Its Simplest Underlying Communication Protocol , 2008, ISAAC.

[6]  Guillaume Theyssier.  Automates cellulaires : un modèle de complexités. (Cellular automata: a model of complexities) , 2005 .

[7]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[8]  John von Neumann,et al.  Theory Of Self Reproducing Automata , 1967 .

[9]  M. Delorme,et al.  Cellular automata : a parallel model , 1999 .

[10]  G. A. Hedlund.  Endomorphisms and automorphisms of the shift dynamical system , 1969, Mathematical systems theory.

[11]  Marvin Minsky,et al.  Computation : finite and infinite machines , 2016 .

[12]  Nicolas Ollinger.  Universalities in cellular automata a (short) survey , 2008, JAC.

[13]  Ludwig Staiger,et al.  Ω-languages , 1997 .

[14]  Ivan Rapaport,et al.  Cellular automata and communication complexity , 2002, Theor. Comput. Sci..

[15]  Juraj Hromkovic,et al.  Communication Complexity and Sequential Compuation , 1997, MFCS.

[16]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[17]  S. Yau.  Mathematics and its applications , 2002 .

[18]  P. Kurka.  Languages, equicontinuity and attractors in cellular automata , 1997, Ergodic Theory and Dynamical Systems.

[19]  Turlough Neary,et al.  P-completeness of Cellular Automaton Rule 110 , 2006, ICALP.