Intrinsic Simulations and Universality in Automata Networks

Martín Ríos-Wilson, Guillaume Theyssier

An automata network (AN) is a finite graph where each node holds a state from a finite alphabet and is equipped with a local map defining the evolution of the state of the node depending on its neighbors. They are studied both from the dynamical and the computational complexity point of view. Inspired from well-established notions in the context of cellular automata, we develop a theory of intrinsic simulations and universality for families of automata networks. We establish many consequences of intrinsic universality in terms of complexity of orbits (periods of attractors, transients, etc) as well as hardness of the standard well-studied decision problems for automata networks (short/long term prediction, reachability, etc). In the way, we prove orthogonality results for these problems: the hardness of a single one does not imply hardness of the others, while intrinsic universality implies hardness of all of them. As a complement, we develop a proof technique to establish intrinsic simulation and universality results which is suitable to deal with families of symmetric networks were connections are non-oriented. It is based on an operation of glueing of networks, which allows to produce complex orbits in large networks from compatible pseudo-orbits in small networks. As an illustration, we give a short proof that the family of networks were each node obeys the rule of the 'game of life' cellular automaton is strongly universal. This formalism and proof technique is also applied in a companion paper devoted to studying the effect of update schedules on intrinsic universality for concrete symmetric families of automata networks.

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