The number of cells that are not white on row t in this case is given by Apply[Times, 1 + IntegerDigits[t, k]] . … Mod[Binomial[t, n], k] is given for prime k by With[{d = Ceiling[Log[k, Max[t, n] + 1]]}, Mod[Apply[Times, Apply[Binomial, Transpose[ {IntegerDigits[t, k, d] , IntegerDigits[n, k, d] }], {1}]], k]] The patterns obtained for any k are nested.

Limited size versions [of multiway systems] One can set up multiway systems of limited size by applying transformations cyclically to strings.

Each picture can be generated by starting from initial conditions at the top, and then just evolving down the page repeatedly applying the cellular automaton rule.

Genetic algorithms As mentioned on page 985 , it is straightforward to apply natural selection to computer programs, and for certain kinds of practical tasks with appropriate continuity properties this may be a useful approach.

In a sequential substitution system only the first replacement that is found to apply in a left-to-right scan is ever performed at any step.

Most of these operations are just done by applying ListConvolve with simple kernels. … Most schemes like this can ultimately be thought of as picking out templates or applying simple cellular automaton rules.

To assess the randomness of a sequence produced by something like a cellular automaton, therefore, what we must do is to apply to it the same methods of analysis as we do to natural systems. … Yet starting with a simple initial condition and then applying a simple cellular automaton rule constitutes a simple

And what this means is that in effect the same rules will apply regardless of how fast one is going. … At the outset it might not have seemed conceivable that any system which at some level just applies a fixed program to various underlying elements could successfully capture the phenomenon of motion.

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.

In each part of the proof each line can be obtained from the previous one just as on page 775 by applying the axiom or lemma indicated.