This led in the 1920s and 1930s to the introduction of various idealizations for mathematics—notably recursive functions, combinators, lambda calculus, string rewriting systems and Turing machines.

With a rule given in this form, each step in the evolution of the mobile automaton corresponds to the function MAStep[rule_, {list_List, n_Integer}] /; (1 < n < Length[list]) := Apply[{ReplacePart[list, #1, n], n + #2}&, Replace[Take[list, {n - 1, n + 1}], rule]] The complete evolution for many steps can then be obtained with MAEvolveList[rule_, init_List, t_Integer] := NestList[MAStep[rule, #]&, init, t] (The program will run more efficiently if Dispatch is applied to the rule before giving it as input.)

But there is evidence that a widespread fractal structure develops—with a correlation function of the form r -1.8 —in the distribution of stars in our galaxy, galaxies in clusters and clusters in superclusters, perhaps suggesting the existence of general overall laws for self-gravitating systems.

The plots below show p[n] and q[n] as a function of n .

General associative [cellular automaton] rules With a cellular automaton rule in which the new color of a cell is given by f[a 1 , a 2 ] (compare page 886 ) it turns out that the pattern generated by evolution from a single non-white cell is always nested if the function f has the property of being associative or Flat .

The locality of cellular automaton rules was thought of as making them the analog for symbol sequences of continuous functions for real numbers (compare page 869 ).

The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]

In the 1990s various idealizations of physics simulations—based on random walks, correlation functions, localization of eigenstates, and so on—were used as tests of pseudorandom generators.

The picture below shows as a function of a the minimum x that solves the equation.

But for standard continuous spaces this definition is hard to make robust—since unlike in discrete networks where one can define volume just by counting nodes, defining volume in a continuous space requires assigning a potentially arbitrary density function.