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Strong stochastic stability of cellular automata

Hugo Marsan, Mathieu Sablik

TL;DR

It is shown that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation by proving that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation.

Abstract

We define the notion of stochastic stability, already present in the literature in the context of smooth dynamical systems, for invariant measures of cellular automata perturbed by a random noise, and the notion of strongly stochastically stable cellular automaton. We study these notions on basic examples (nilpotent cellular automata, spreading symbols) using different methods inspired by those presented in \cite{MST19}. We then show that this notion of stability is not trivial by proving that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation.

Strong stochastic stability of cellular automata

TL;DR

It is shown that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation by proving that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation.

Abstract

We define the notion of stochastic stability, already present in the literature in the context of smooth dynamical systems, for invariant measures of cellular automata perturbed by a random noise, and the notion of strongly stochastically stable cellular automaton. We study these notions on basic examples (nilpotent cellular automata, spreading symbols) using different methods inspired by those presented in \cite{MST19}. We then show that this notion of stability is not trivial by proving that a Turing machine cannot decide if a given invariant measure of a cellular automaton is stable under a uniform perturbation.
Paper Structure (27 sections, 15 theorems, 36 equations, 9 figures, 1 table)

This paper contains 27 sections, 15 theorems, 36 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stochastic stability: physics motivation
  3. Cellular automata: computer science (and other) motivation
  4. Stochastic stability for cellular automata
  5. Description of the paper
  6. Stochastic stability for cellular automata
  7. Nilpotent CA
  8. Spreading CA
  9. 1-dimensional
  10. Spread graph
  11. Update functions
  12. Proof of the theorem
  13. $d$-dimensional binary
  14. Computational lemmas
  15. Undecidability
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $\left(F_{\epsilon}\right)_{\epsilon>0}$ a family of $\epsilon$-perturbations of a nilpotent CA $F$ on $\mathbb{Z}^{d}$. For $\epsilon$ small enough, we denote by $\pi_{\epsilon}$ the unique invariant measure of $F_{\epsilon}$. Then there is a constant $C>0$ such that for all finite $A\subset\ma

Figures (9)

  • Figure 3.1: Proof of theorem \ref{['thm:Nilpotents']}. To have a $0$ at $t=0$, it suffices to not make any mistake on the cells inside the grayed area.
  • Figure 4.1: Left: step 1, the first three levels of the initial dependency graph for $r=3$. Center: step 2, the decomposition into a planar graph. Right: step 3, adding the noise.
  • Figure 4.2: Left: The first three levels of the original graph. Right: the first three levels of the the dual graph. Note that the outer vertices actually represent the same region of the original graph.
  • Figure 4.3: Proof of theorem \ref{['thm:Spreading-general']}. To have a $0$ when $t=0$ on all $A$, it suffices to have one in $\left(-t_{\left|A\right|},a\right)$ and not make any mistakes on the cells of the colored area. Time goes upward.
  • Figure 4.4: Illustration for neighborhood $\mathcal{N}=\{0,1\}$. In blue the spread graph, in black the dual graph. Vertical red arrows have a probability $\epsilon$ to be “ closed” , and thus the horizontal dashed red ones have a probability $\epsilon$ to be “ open” .
  • ...and 4 more figures

Theorems & Definitions (37)

  • Definition 2.1: Stochastic stability of a measure
  • Definition 2.2: Strong stochastic stability of a cellular automaton
  • Theorem 3.1: Stability for nilpotent CA
  • proof
  • Example 4.1
  • Theorem 4.2
  • Definition 4.3
  • Lemma 4.4: FT08
  • Corollary 4.5
  • proof
  • ...and 27 more