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Communication complexity meets cellular automata: Necessary conditions for intrinsic universality

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Abstract

A natural way to interpret a cellular automaton (CA) is as a mechanism that computes, in a distributed way, some function f. In other words, from a computer science point of view, CAs can be seen as distributed systems where the cells of the CAs are nodes of a network linked by communication channels. A classic measure of efficiency in such distributed systems is the number of bits exchanged during the computation process. A typical approach is to look for bottlenecks: channels through which the nature of the function f forces the exchange of a significant number of bits. In practice, a widely used way to understand such congestion phenomena is to partition the system into two subsystems and try to find bounds for the number of bits that must pass through the channels that join them. Finding these bounds is the focus of communication complexity theory. Measuring the communication complexity of some problems induced by a CA \(\phi\) turned out to be tremendously useful to give clues regarding the intrinsic universality of \(\phi\) (a CA is said to be intrinsically universal if it is capable of emulating any other). In fact, there exist particular problems \(\mathrm {P}\)’s for which the following key property holds: if \(\phi\) is intrinsically universal, then the communication complexity of \(\mathrm {P}(\phi )\) must be maximal. In this tutorial, we intend to explain the connections that were found, through a series of papers, between intrinsic universality and communication complexity in CAs.

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Acknowledgements

The first author was supported by ANID/FONDECYT de Iniciación en Investigación 11200892. The second author was supported by ANID/PIA Apoyo a Centros Científicos y Tecnológicos de Excelencia AFB 170001 and Fondecyt 1170021.

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Raimundo Briceño

  2. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile

    Ivan Rapaport

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  1. Raimundo Briceño
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  2. Ivan Rapaport
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Correspondence to Ivan Rapaport.

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Briceño, R., Rapaport, I. Communication complexity meets cellular automata: Necessary conditions for intrinsic universality. Nat Comput 20, 307–320 (2021). https://doi.org/10.1007/s11047-021-09857-z

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