I strongly suspect that it is true in general that any cellular automaton which shows overall class 4 behavior will turn out—like rule 110—to be universal.

Knowing that universal systems exist already tells one that this must be true at least in some situations.

And what is more important, it turns out that when there is true computational reducibility its effect is usually much more dramatic.

But at some level the same is probably true of the individual nerve cells in our brains.

But particularly after my discoveries in Chapter 9 , I strongly suspect that even if this is formally the case, it will still not turn out to be a true representation of ultimate physical reality, but will instead just be found to reflect various idealizations made in the models used so far.

But this all changed in 1931 when Gödel's Theorem showed that at least in any finitely-specified axiom system containing standard arithmetic there must inevitably be statements that cannot be proved either true or false using the rules of the axiom system.

But the Principle of Computational Equivalence also implies that the same is ultimately true of our whole universe.

In general, the more systematic the proofs in a particular area become, the less relevant they will typically seem compared to the theorems that they establish as true.

Given the original list s and the complete prime implicant list p the so-called Quine–McCluskey procedure can be used to find a minimal list of prime implicants, and thus a minimal DNF: QM[s_, p_] := First[Sort[Map[p 〚 # 〛 &, h[{}, Range[Length[s]], Outer[MatchQ, s, p, 1]]]]] h[i_, r_, t_] := Flatten[Map[h[Join[i, r 〚 # 〛 ], Drop[r, #], Delete[Drop[t, {}, #], Position[t 〚 All, # 〛 ], {True}]]] &, First[Sort[Position[#, True] &, t]]]], 1] h[i_, _, {}] := {i} The number of steps required in this procedure can increase exponentially with the length of p .

Sometimes clarity demands that I say explicitly that something from the past is wrong, but generally I try to avoid this, preferring instead just to state whatever I now believe is true.