Understanding a Non-trivial Cellular Automaton by Finding Its Simplest Underlying Communication Protocol

  • Authors:

  • Eric Goles;Cedric Little;Ivan Rapaport

  • Affiliations:

  • Facultad de Ciencias y Tecnologías, Universidad Adolfo Ibáñez, Chile and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile,;Facultad de Ciencias y Tecnologías, Universidad Adolfo Ibáñez, Chile;Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, and Departamento de Ingeniería Matemática, Universidad de Chile,

  • Venue:
  • ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
  • Year:
  • 2008

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Abstract

In the present work we find a non-trivial communicationprotocol describing the dynamics of an elementary CA, and we provethat there are no simpler descriptions (protocols) forsuch CA. This is, to our knowledge, the first time such a result isobtained in the study of CAs. More precisely, we divide the cellsof Rule 218 into two groups and we describe (and thereforeunderstand) its global dynamics by finding a protocol taking placebetween these two parts. We assume that x ε{0,1}n is given to Alice while yε {0,1}n is given to Bob. Let us callz(x,y) ε {0,1} the result of thedynamical interaction between the cells. We exhibit a protocolwhere Alice, instead of the n bits of x, sends2[log(n)] + 1 bits to Bob allowing him to computez(x,y). Roughly, she sends 2 particularpositions of her string x. By proving that any one-roundprotocol computing z(x,y) must exchangeat least 2[log(n)]- 5 bits, the optimality of ourconstruction (up to a constant) is concluded.