Encodings [for universality] One can prevent an encoding from itself introducing universality by insisting, for example, that it be primitive recursive (see page 907 ) or always involve only a bounded number of steps.

In the past, people have tended to consider it more scientific to give only numerical summaries of such data.

The initial condition {1, 0, 1} with all cells 0 on the previous step yields a structure that repeats but only every 666 steps. The initial condition {{0, 1, 1}, {1, 0, 0}} yields a pattern that grows sporadically for 3774 steps, then breaks into two repetitive structures.

But if only u is known, then the equations can instead be thought of as providing implicit constraints for v . … But as soon as the original equation is nonlinear, say u  m 1 . v + m 2 . v 2 , the situation changes dramatically. It still takes only about n 2 steps to compute u given v , but it becomes vastly more difficult to compute v given u , taking perhaps 2 2 n steps.

In fact, it turns out that in continuous cellular automata it takes only extremely simple rules to generate behavior of considerable complexity.

But as soon as one perturbs such initial conditions, one normally seems to get only complicated and seemingly random behavior, as in the top row of pictures in the second image.

But complete evidence that human thinking follows the Principle of Computational Equivalence will presumably come only gradually as practical computer systems manage to emulate more and more aspects of human thinking.

Statements that can be proved with induction but are not provable only with Robinson's axioms are: x ≠ Δ x ; x + y  y + x ; x + (y + z)  (x + y) + z ; 0 + x  x ; ∃ x ( Δ x + y  z ⇒ y ≠ z) ; x × y  y × x ; x × (y × z)  (x × y) × z ; x × (y + z)  x × y + x × z .

{x, y}, 1] where m is a matrix such as {{2, 1}, {1, 1}} . Any initial condition containing only rational numbers will then yield repetitive behavior, much as in the shift map.

Recognizing repetition [in sounds] The curve of the function Sin[x] + Sin[ √ 2 x] shown on page 146 looks complicated to the eye. … And this fact is presumably why musical scores normally have notes only at integer multiples of some fixed time interval.