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Descriptive Complexity of Sensitivity of Cellular Automata

Tom Favereau, Ville Salo

TL;DR

This work shows that the computational complexity of determining whether a cellular automaton is sensitive to initial conditions is $\Pi^0_2$-complete in dimension 1 and $\Sigma^0_3$-complete in dimension 2 and higher.

Abstract

We study the computational complexity of determining whether a cellular automaton is sensitive to initial conditions. We show that this problem is $Π^0_2$-complete in dimension 1 and $Σ^0_3$-complete in dimension 2 and higher. This solves a question posed by Sablik and Theyssier.

Descriptive Complexity of Sensitivity of Cellular Automata

TL;DR

This work shows that the computational complexity of determining whether a cellular automaton is sensitive to initial conditions is -complete in dimension 1 and -complete in dimension 2 and higher.

Abstract

We study the computational complexity of determining whether a cellular automaton is sensitive to initial conditions. We show that this problem is -complete in dimension 1 and -complete in dimension 2 and higher. This solves a question posed by Sablik and Theyssier.

Paper Structure

This paper contains 15 sections, 17 theorems, 9 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-dimensional Cellular Automata
  4. Upper Bound on Complexity of the Sensitivity Problem
  5. Construction of $G_e$
  6. Lower Bound and $\Pi^0_2$-Hardness
  7. If $e \in \text{TOT}$:
  8. If $e \notin \text{TOT}$:
  9. Higher Dimensional Cellular Automata
  10. Upper Bound: $\Sigma^0_3$
  11. Lower Bound: $\Sigma^0_3$-Hardness
  12. If $e \in \text{COF}$:
  13. If $e \notin \text{COF}$:
  14. Application
  15. Conclusion and Future Work

Key Result

theorem thmcountertheorem

The problem of determining whether a one-dimensional cellular automaton is sensitive is $\Pi^0_2$-complete.

Figures (4)

  • Figure 1: Structure of computational blocks in $G_e$. The upper row shows the delimiters and machine heads, while the lower rows show the tape content.
  • Figure 2: Complete transition function of the cellular automaton $G_e$. Black cells act as wildcards and can match any state in the neighborhood. When no rule applies, the cell remains unchanged.
  • Figure 3: Main rules for the transition function of the 2-dimensional cellular automaton $G_e$. Black cells act as wildcards and can match any state in the neighborhood. All rules specify the next state of the center cell, except for the last one, which applies to a full neighborhood configuration (but remains a local rule). When no rule applies, the cell remains unchanged.
  • Figure 4: Illustration of the stabilization process. (a) Defective red regions are progressively eliminated. (b) Green regions are removed in parallel. (c) A representative configuration after the system has stabilized.

Theorems & Definitions (39)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • definition thmcounterdefinition: Cellular Automaton
  • definition thmcounterdefinition: Sensitivity
  • definition thmcounterdefinition: m-blocking word
  • theorem thmcountertheorem: Kůrka kurka2008
  • definition thmcounterdefinition: Higher dimensional m-blocking word
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 29 more