A proof of the equivalence (p ⊼ q)  (q ⊼ p) between logic expressions is then formed by applying these axioms in the particular sequence shown.

One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!Array[Apply[Times, Map[(1 - Mod[#, 2])(# - 1)!!

., 0}] := -Apply[Plus, 4 Range[Length[{p}]] - 1] + 6 d[{__, 1, p : 0 .., 0}] := d[{1, p, 0}] - 7 d[{___, p : 1 .., q : 0 ..., 1, 0}] := 4 Length[{p}] + 3 Length[{q}] + 2 d[{___, p : 1 .., 1, 0}] := 4 Length[{p}] + 2

But in general it may be undecidable even whether two such forms are actually equivalent (compare the notes below and on page 1051 )—since to tell this one might need to be able to apply the rules infinitely many times.

But to apply such an argument one must among other things assume that we can imagine all the ways in which intelligence could conceivably operate.

The transitions between these states have probabilities given by m[Map[Length, list]] where m[s_] := With[{q = FoldList[Plus, 0, s]}, ReplacePart[ RotateRight[IdentityMatrix[Last[q]], {0, 1}], 1/Length[s], Flatten[Outer[List, Rest[q], Drop[q, -1] + 1], 1]]] The average spectrum of sequences generated according to these probabilities can be obtained by computing the correlation function for elements a distance r apart ξ [list_, r_] := With[{w = (# - Apply[Plus, #]/Length[#] &)[ Flatten[list]]}, w . … The same basic setup also applies to spectra associated with linear filters and ARMA time series processes (see page 1083 ), in which elements in a sequence are generated from external random noise by forming linear combinations of the noise with definite configurations of elements in the sequence.

Applying FoldList[Plus, 0, 2list - 1] to the whole sequence yields the pattern shown below. … This is similar to picture (c) on page 131 , and is a digit-by-digit version of FoldList[Plus, 0, Table[Apply[Plus, 2 Rest[IntegerDigits[i, 2]] - 1], {i, n}]] Note that although the picture above has a nested structure, the original concatenation sequences are not nested, and so cannot be generated by substitution systems.

In a system like a cellular automaton that is based on explicit rules, it is always straightforward to take the rule and apply it to see Examples of patterns produced by systems in which not only must the arrangement of colors in each neighborhood match one of a fixed set of templates, but also a certain template from this set must occur at least once in the pattern.

And by thinking in terms of such computations, it then becomes possible to imagine formulating principles that apply to a very wide variety of different systems—quite independent of the detailed structure of their underlying rules.

Or does the notion of computation somehow apply only to systems with abstract elements like, say, the black and white cells in a cellular automaton?