But the discovery of computational irreducibility now implies that this can fundamentally never happen, and that in fact there can be no easy theory for almost any behavior that seems to us complex.

And among other things this then implies many fundamental limits on the processes of perception and analysis that we discussed in Chapter 10 .

The operators shown are And , Equal , Implies and Nand .

And indeed, whenever one is dealing with theorems that can be derived from a finite axiom system the criterion implies that only a finite number of theorems should ever be considered interesting—ending as soon as one has in a sense got enough theorems to be able to reproduce some formulation of the axiom system.

As soon as there are a limited number of possible keys then in principle one can always try each of them in turn, looking in each case to see whether they imply an original message that is meaningful in the language in which the message is written.

The mechanics of the ear imply that if two tones of reasonable amplitude are played together, progressively smaller additional signals will effectively be generated at frequencies Abs[n 1 ω 1 ± n 2 ω 2 ] .

In quantum theory the quantization of particle spin implies that any photon hitting a polarizing filter will always either just go through or be absorbed—so that in effect its spin measured relative to the orientation of the polarizer is either +1 or -1. … In ordinary quantum theory, a straightforward calculation implies that the expected value of the product of the two measured spin values will be -Cos[ θ ] . … But for triples of particles, it was noticed in the late 1980s that they can give constraints that force probabilities to be 0 or 1, implying that with the assumptions made, certain configurations of measurement results are simply impossible.

The universality of rule 110 presumably implies that the axiom system given is universal.

And this immediately implies that the pattern must always have a nested form.

But an experiment which failed to see any of these particles implied that the rate must be much lower.