simple models do not necessarily have simple behavior. … In real markets, it is usually impossible to see in detail what each entity is doing. … But although this will make it more difficult to recognize definite rules even if one looks at the complete behavior of every element in the system, it does not affect the basic point that there is randomness that can intrinsically be generated by the evolution of the system.

So this means that given a sufficiently precise knowledge of the state of a physical system at the present time, it is therefore possible to deduce not only what the system will do in the future, but also what it did in the past. In the first cellular automaton shown below it is also straightforward to do this. … But the second cellular automaton works differently, and does not allow one to go backwards.

The second set of pictures below demonstrates why two-dimensional block encoding does, however, manage to compress it. … In cases (e) and (f), however, there is no simple rule for going from one row to the next, and two-dimensional block encoding—like all the other encoding schemes we have discussed so far—does not yield any substantial compression. … And although such a procedure does not in the Examples of two-dimensional block-based encoding.

So are there other ways to do the same computation in a different number of steps? … It turns out that there are 351 different functions that can be computed by one or more of the 4096 Turing machines with 2 states Examples of the behavior of a simple Turing machine that does the computation of adding 1 to a number. … The average for a given length of input does not increase with n , and is always precisely 5.

But even though the underlying rules increase rapidly in complexity, the overall forms of behavior that we see do not change much. … Allowing four or more colors, however, does not further increase the complexity of the behavior, and, as the picture shows, even with five colors, simple repetitive and nested behavior can still occur.

Properties [of example Turing machines] The maximum numbers of steps increase with input size according to: (a) 14 2^Floor[n/2] - 11 + 2Mod[n, 2] (b) (does not halt for x = 1 ) (c) 2 n - 1 (d) (7(1 + Mod[n, 2])4^Floor[n/2] + 2Mod[n, 2] - 7)/3 (h) (see note below) (i) (does not halt for various x > 53 ) (j) (does not halt for various x > 39 ) (k) (does not halt for x = 1 ) (l) 5 (2 n - 2 - 1)

So what kinds of processes does such analysis involve? … Most kinds of statistical analysis are fundamentally based on the assumption that such models must be probabilistic, in the sense that they give only probabilities for behavior, and do not specifically say what the behavior will be. … If one has a deterministic model then it is at least in principle quite straightforward to find out whether the model is correct: for all one has to do is to compare whatever specific behavior the model predicts with behavior that one observes.

But it is rather easy to foil this particular approach to cryptanalysis: all one need do is not sample every single cell in a given column in forming the encrypting sequence. For without every cell there does not appear to be enough information for any kind of local rule to be able to deduce one column from others. … And since rule 30 is not additive, it simply does not work.

For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants. … What one does is to look at the digit sequences for the numbers that give the vertical and horizontal positions of a certain square. … So why does this procedure work?

And if one does this, one immediately gets all sorts of fairly complicated patterns that are often not just purely nested—as illustrated in the pictures on the next page . … The presence of this nested structure is an inevitable consequence of the fact that the rule for replacing an element at a particular position does not depend in any way on other elements.