A Graph Theory Approach for Regional Controllability of Boolean Cellular Automata

TL;DR

The paper addresses regional controllability of Boolean cellular automata under boundary control on a subregion $\omega$. It develops a graph-theoretic framework using a transition graph $\Upsilon$ and a transition matrix $\mathcal{C}$ to model one-step evolutions, establishing two main results: regional controllability is characterized by the existence of a Hamiltonian circuit in $\mathcal{C}^t$ and, more practically, by the graph's strongly connected components. A polynomial-time SCC criterion provides a decidability test, and a preimage-based method constructs explicit boundary controls via a distance function $\Delta_i$ and backtracking paths. The approach yields constructive procedures for 1D and 2D CA, with examples on Wolfram rules illustrating when regional controllability holds and how controls can be synthesized, offering a pathway for boundary-driven steering of subregions in spatially extended discrete systems.

Abstract

Controllability is one of the central concepts of modern control theory that allows a good understanding of a system's behaviour. It consists in constraining a system to reach the desired state from an initial state within a given time interval. When the desired objective affects only a sub-region of the domain, the control is said to be regional. The purpose of this paper is to study a particular case of regional control using cellular automata models since they are spatially extended systems where spatial properties can be easily defined thanks to their intrinsic locality. We investigate the case of boundary controls on the target region using an original approach based on graph theory. Necessary and sufficient conditions are given based on the Hamiltonian Circuit and strongly connected component. The controls are obtained using a preimage approach.

Figures (11)

• Figure 1: Regional control of one-dimensional CA.
• Figure 2: The evolution of CA Wolfram rule 90 on the region $\omega = \{c_{1}, \cdots, c_{6}\}$ starting with the same initial configuration; on the left without control and on the right with control.
• Figure 3: Regional Control of two-dimensional CA
• Figure 4: Evolution of the CA rule example \ref{['Rule2D']} on $\omega$ in the autonomous and controlled cases on the left and right matrices respectively.
• Figure 5: Regional control of two dimensional CA with asymmetric controls

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Example 3.2
• Remark 1
• Example 3.3
• Remark 2
• Remark 3
• Example 4.1