To give evidence that this is not merely a reflection of continual injection of randomness from the environment what is normally done is to show that at least some aspect of the behavior of the system can be fit by a definite simple iterated map or differential equation. But inevitably the fit will only be approximate, so there will always be room for effects from randomness in the environment.

And those with mathematical backgrounds tend to add features to make their systems fit in with complicated and abstract ideas—often related to continuity—that exist in modern mathematics.

In the case shown here, all the replacements found to fit in a left-to-right scan are carried out at each step.

But in fact there are always many somewhat arbitrary details, particularly centering around exactly how to prune less fit organisms.

Surely we cannot simply search through possible rules of certain kinds, looking for one whose behavior happens to fit what we see in physics?

From the names of concepts and people that I mention, it is straightforward to do web or database searches that give a vastly more complete picture of available references than could possibly fit in a book of manageable size—or than could be created correctly without immense scholarship.

Rather, the purpose is usually just A comparison between data generated by ordinary cellular automata and the probabilistic cellular automata that are considered the best fit to it.

Things appear somewhat simpler with boiling points, and as noticed by Harry Wiener in 1947 (and increasingly discussed since the 1970s) these tend to be well fit as being linearly proportional to the so-called topological index given by the sum of the smallest numbers of connections visited in getting between all pairs of carbon atoms in an alkane molecule.

Substitution systems in which all replacements are done that are found to fit in a left-to-right scan can be implemented as follows GSSEvolveList[rule_, s_, n_] := NestList[GSSStep[rule, #] &, s, n] GSSStep[rule_, s_] := g[rule, s, f[StringPosition[s, Map[First, rule]]]] f[{ }] = { }; f[s_] := Fold[If[Last[Last[#1]] ≥ First[#2], #1, Append[#1, #2]]&, {First[s]}, Rest[s]] g[rule_, s_, { }] := s; g[rule_, s_, pos_] := StringReplacePart[ s, Map[StringTake[s, #] &, pos] /. rule, pos] with rules given as {"ABA"  "BAAB", "BBBB"  "AA"} .

(Least squares fits do this for models in which the data exhibits independent Gaussian variations.)