Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Characterization of regional controllability and controllability of deterministic cellular automata via topological and symbolic dynamics notions

Sara Dridi

TL;DR

The paper develops a topological and symbolic-dynamics framework to characterize controllability and regional controllability of one-dimensional deterministic Boolean CA, avoiding the Kalman condition. It introduces a transition-graph criterion showing regional controllability is equivalent to a single SCC in $G_{n}(F)$ and links this to transitivity/mixing properties of the associated SFTs $S_{n}(F)$ and their traces. It further establishes that regional controllability for every region size $n$ corresponds to chain-transitivity (and chain-mixing) and to transitive (and mixing) SFTs, with the 2-approximation trace offering an alternative characterization. The work also identifies blocking words as a fundamental obstruction to controllability, tying dynamical obstructions to control limitations and ultimately showing controllability for CA is equivalent to regional controllability across all scales.

Abstract

This article presents a new characterization of controllability and regional controllability of Deterministic Cellular Automata (CA for short). It focuses on analyzing these problems within the framework of control theory, which have been extensively studied for continuous systems modeled by partial differential equations (PDEs). In the analysis of linear systems, the Kalman rank condition is ubiquitous and has been used to obtain the main results characterizing controllability. The aim of this paper is to highlight new ways to prove the regional controllability and controllability of CA using concepts from symbolic dynamics instead of using the Kalman condition. Necessary and sufficient conditions are given using the notions of chain transitive, chain mixing, trace approximation, transitive SFT and mixing SFT. Finally, we demonstrate that the presence of visibly blocking words implies that the cellular automaton is not controllable.

New Characterization of regional controllability and controllability of deterministic cellular automata via topological and symbolic dynamics notions

TL;DR

The paper develops a topological and symbolic-dynamics framework to characterize controllability and regional controllability of one-dimensional deterministic Boolean CA, avoiding the Kalman condition. It introduces a transition-graph criterion showing regional controllability is equivalent to a single SCC in and links this to transitivity/mixing properties of the associated SFTs and their traces. It further establishes that regional controllability for every region size corresponds to chain-transitivity (and chain-mixing) and to transitive (and mixing) SFTs, with the 2-approximation trace offering an alternative characterization. The work also identifies blocking words as a fundamental obstruction to controllability, tying dynamical obstructions to control limitations and ultimately showing controllability for CA is equivalent to regional controllability across all scales.

Abstract

This article presents a new characterization of controllability and regional controllability of Deterministic Cellular Automata (CA for short). It focuses on analyzing these problems within the framework of control theory, which have been extensively studied for continuous systems modeled by partial differential equations (PDEs). In the analysis of linear systems, the Kalman rank condition is ubiquitous and has been used to obtain the main results characterizing controllability. The aim of this paper is to highlight new ways to prove the regional controllability and controllability of CA using concepts from symbolic dynamics instead of using the Kalman condition. Necessary and sufficient conditions are given using the notions of chain transitive, chain mixing, trace approximation, transitive SFT and mixing SFT. Finally, we demonstrate that the presence of visibly blocking words implies that the cellular automaton is not controllable.
Paper Structure (13 sections, 18 theorems, 18 equations, 5 figures)

This paper contains 13 sections, 18 theorems, 18 equations, 5 figures.

Table of Contents

  1. DEFINITIONS AND BACKGROUND
  2. Topological dynamics
  3. Symbolic dynamics
  4. Cellular automata
  5. Problems statement
  6. Regional Controllability of Cellular automata
  7. Controllability of Cellular automata
  8. Characterizing regional controllability for Boolean deterministic CA using notions of symbolic dynamics
  9. Transition graph approach and regional controllability problem
  10. Necessary and Sufficient Condition: Chain transitive and Chain mixing
  11. Necessary and Sufficient Condition: Transitive SFT and Mixing SFT
  12. Necessary and Sufficient Condition: Trace and Approximation Trace
  13. Characterizing Controllability for Boolean deterministic CA using notions of symbolic dynamics

Key Result

Theorem 3.1

dridi2019graph A CA $(A^{\mathbb{Z}},F)$ is regionally controllable for a given rule iff the transition graph $G_{n}(F)$ associated to the rule has only one Strongly Connected Component (SCC).

Figures (5)

  • Figure 1: Space-time diagram and trace over segment $[-1,1]$
  • Figure 2: Regional controllability of one dimensional cellular automaton
  • Figure 3: Controllability of one dimensional cellular automaton
  • Figure 4: Evolution of Wolfram Rule $90$ Cellular Automaton on the Region $\omega=\{c_{1},\dots,c_{6}\}$ Beginning with the same initial configuration: uncontrolled on the left and controlled on the right
  • Figure 5: A blocking word

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Example 2.1
  • Definition 7
  • Example 2.2
  • Theorem 3.1
  • ...and 40 more