Writing in 1993, a researcher noted that it is hard to prove things about a cellular automata model - has this changed?

Question

Leah Edelstein-Keshet in her 1993 article Cellular automata approaches to biological modelling writes:

We do not believe that CA should be viewed as a replacement for rigorous mathematical models. Once it has been established that a CA implementation of one's hypothesis produces the desired results, then one must proceed towards deriving a traditional mathematical model. For then and only then is it possible to bring to bear tools from analysis such as stability theory, bifurcation theory and analysis theory. Only in very rare cases has it been possible to prove the existence of some particular behaviour for a CA.

How has our repertoire of tools designed to prove things about a discrete set of rules, such as cellular automata, changed since then? Does simulation remain our only approach?

Answer 1

• opposing CA to "rigorous mathematical models" and then complaining that proving the existence of some behavior is almost impossible for CA is somewhat funny: first, CA are perfectly rigorous mathematical models and, second, the classical approach by PDE leads to similar difficulties (even the existence of a solution can be challenging).

• like for the "classical" approach by PDE, the linear case is easier for cellular automata: their ergodic dynamics is reasonably well-understood and they can be predicted quickly (i.e. faster than simulation). This predictability also extends to some non-linear cases arxiv:patt-sol/9701008.

• concerning the evolution of the "repertoire of tools" to analyze CA since 1993, maybe we can cite the particle/collision approach and the breakthrough results about rule 110, and also the progress made on "number conserving cellular automata" which are particularly relevant for modeling natural phenomena.