And what we see is that in the first three pictures, there are many obvious such deviations, while in the remaining pictures there are no obvious deviations. So from this it is fairly easy to conclude that the first three sequences are definitely not random, while the remaining sequences could still be random.

Looking at the specific universal cellular automaton that we have discussed in this section , however, we would probably be led to assume that while the phenomenon of universality might be important in principle, it would rarely be relevant in practice. … The main difference between a mobile automaton and a cellular automaton is that in a mobile automaton there is a special active cell that moves around from one step to the next, while in a cellular

Autoplectic processes In the 1985 paper where I introduced intrinsic randomness generation I called processes that show this autoplectic, while I called processes that transcribe randomness from outside homoplectic.

In rule (f) the causal network grows exponentially, while in rule (e) the causal network also grows quite rapidly, though its overall growth properties are not clear.

And looking at the plots of sizes of numbers produced, one sees that for quite a while these two different initial conditions lead to results that are indistinguishably close. … And for a while these digits are

And while a particular machine may be able to control the initial position of a point to a certain accuracy, such repeated amplification will eventually lead to sensitivity to still smaller changes. … But after a while, they reach the edge of the material and cannot diverge any

And while none of these yet look complicated enough that they might reasonably be called random, I suspect that in time similar but vastly more complex examples will be found. … But while some of the simpler ones have been captured quite completely by methods based on traditional mathematical equations, the more complex ones have not.

But what is somewhat special about the setup above is that inputs which yield the same output tend to be ones that might reasonably be considered similar, while inputs that yield different outputs tend to be significantly different. And thus, for example, a change in a single input cell typically will not have a high probability of affecting the output, while a change in a large fraction of the input cells will.

And sometimes the proof may be straightforward—say being based on showing that one side of the equation is always odd while the other is always even. … And the way this works is that for example one variable in the equation gives the number of steps of evolution, while another gives the outcome after that number of steps.

And indeed, while operator systems and multiway systems have many superficial differences, I suspect that when it comes to universality they work very much the same. … For while it is common, say, to take a problem in geometry and reformulate it as a problem in algebra, this is almost always done just by setting up a direct