Cellular automaton model of self-healing

Abstract

We propose a simple cellular automaton model of a self-healing system and investigate its properties. In the model, the substrate is a two-dimensional checkerboard configuration which can be damaged by changing values of a finite number of sites. The cellular automaton we consider is a checkerboard voting rule, a binary rule with Moore neighbourhood which is topologically conjugate to majority voting rule. For a single color damage (when only cells in the same state are modified), the rule always fixes the damage. For a general damage, when it is localized inside a $3\times 3$ square, the rule also fixes it always. When the damage is inside of a larger $n\times n$ square, the efficiency of the rule in fixing the damage becomes smaller than $100\%$, but it remains better than $98\%$ for $n\le 5$ and better than $75\%$ for $n\le 7$. We show that in the limit of infinite $n$ the efficiency tends to zero.

12 pages, 5 figures