Motivation / discussion: Conway's Game of Life automaton is often suggested as a model for either real (biological) life and/or computation, but one important trend in both real life and computer systems that I've never seen discussed in the context of the Game of Life (=: GoL) is error correction, meaning the ability to survive/correct small errors. On the contrary, GoL configurations tend to be extremely brittle, vulnerable to even the tiniest change. My question is basically whether there are “robust” configurations in the GoL in the sense that they can resist small errors — and how we can build them. The question is somewhat similar in spirit to this earlier one by Joel D. Hamkins about “superstable” configurations but differs in that Joel's question is about surrounding a configuration by an arbitrary (hostile) environment, whereas mine is about making (small) changes to the configuration itself.
Definition: Consider first the situation of the Game of Life on a finite (flat) torus $(\mathbb{Z}/n_1\mathbb{Z}) \times (\mathbb{Z}/n_2\mathbb{Z})$ or some other kind of bounded domain (possibly with edge conditions) admitting only finitely many configurations: if $k \in \mathbb{N}$, say that a configuration $C$ is $k$-robust when any configuration $C'$ of Hamming distance $\leq k$ to $C$ (i.e., differing in the state of at most $k$ cells) will evolve identically to $C$ beyond a certain time, i.e., there is $N$ such that the evolution of $C$ and $C'$ coincide after $N$ steps (and thenceforth). Since I considered a bounded domain, of course, it doesn't matter whether the $\exists N$ is placed before or after the $\forall C'$ in this statement.
Clearly, at least $2$-robust configurations exist, since the all-dead one is such (any configuration with $\leq 2$ living cells evolves to the all-dead one).
Generalizing to the game of life on $\mathbb{Z}^2$ (where we must make allowance for the possible existence of gliders), say that a configuration $C$ is $k$-robust when any configuration $C'$ of Hamming distance $\leq k$ to $C$ will evolve locally identically to $C$ in the sense that for any bounded (i.e., finite) domain $B \subseteq \mathbb{Z}^2$ there exists $N$ such that the evolution of $C$ and $C'$ coincide on $B$ after $N$ steps and thenceforth. To make the quantifiers clear:
- $C$ is $k$-robust iff $\forall C'$ such that $\operatorname{dist}(C,C') \leq k$ and $\forall B \subseteq \mathbb{Z}^2$ finite, $\exists N$ such that $\forall i\geq N$, $\operatorname{evol}^i(C')|_B = \operatorname{evol}^i(C)|_B$.
Question: Do there exist $k$-robust configurations for all $k$?
Note: Since the question is probably difficult, an answer with any reasonable modification of the definition of “robust” will be accepted, or reasonable modification of the setup (e.g., if for all $k$ there exists arbitrarily large torus sizes with $k$-robust configurations on the torus, that's a fine answer).
