And a major apparent problem was that if everything—including the measuring device—is supposed to be treated as part of the same quantum system, then all of it must follow the rules for pure quantum processes, which do not explicitly include any reduction of the kind supposed to occur in observations. … But this does not mean that in more complicated systems more characteristic of real measuring devices there may not be other sources of randomness that end up dominating. … Curiously, what the Principle of Computational Equivalence suggests is that when quantum systems intrinsically produce apparent randomness they will in the end typically be capable of doing computations just as sophisticated as any other system—and in particular just as sophisticated as would be involved in conscious perception.

Pointer-based encoding One can encode a list of data d by generating pointers to the longest and most recent copies of each subsequence of length at least b using PEncode[d_, b_ : 4] := Module[{i, a, u, v}, i = 2; a = {First[d]}; While[i ≤ Length[d], {u, v} = Last[Sort[Table[{MatchLength[d, i, j], j}, {j, i - 1}]]]; If[u ≥ b, AppendTo[a, p[i - v, u]]; i += u, AppendTo[a, d 〚 i 〛 ]; i++]]; a] MatchLength[d_, i_, j_] := With[{m = Length[d] - i}, Catch[ Do[If[d 〚 i + k 〛 =!

What I do in this book goes significantly further than traditional science in getting rid of notions of purpose from investigations of nature.

(A way to do this for pairs of non-negative integers is to use σ [{x_, y_}] := 1/2(x + y)(x + y + 1) + x .)

And to do this approximately is not difficult.

(As discussed on page 1070 , this representation is unique so long as one does not allow any pairs of adjacent 1's in the digit sequence.)

A triangular lattice is one example where they do not.

But the spatial and temporal entropies that I introduced do not—and indeed in studying specific cellular automata there seems to be no particular reason why such a property would be useful.

Grand unified models typically do this for all known gauge bosons (except gravitons) and for corresponding families of quarks and leptons—and inevitably imply the existence of various additional particles more massive than those known, but with properties that are somehow intermediate.

Page 1186 gives some results, and suggests that sequences which require more complicated cellular automaton rules do tend to look to us more complicated and more random.