[No text on this page] Emulating a Turing machine with a tag system that depends only on the first element at each step. The configuration of cells on each side of the head in the Turing machine is treated as a base 2 number.

But the crucial point is that the behavior we see will only ever be as random as the sequence of digits in the initial conditions. … And ultimately this question can only be answered by going outside of the system one is looking at, and studying whatever it was that set up its initial conditions. … The basis for this is the traditional mathematical idealization that the only relevant attribute of any number is its size.

But the key to understanding what is going on is simply to realize that one has to think not only about the systems one is studying, but also about the types of experiments and observations that one uses in the process of studying them. The crucial point then turns out to be that practical experiments almost inevitably end up involving only initial conditions that are fairly simple for us to describe and construct. … The consequence of this is that no reasonable experiment can ever involve setting up the kind of initial conditions that will lead to decreases in randomness, and that therefore all practical experiments will tend to show only increases in randomness.

The basic idea is to have a sequence of layers of nerve cells—much as one knows exist in the brain—with each cell in each successive layer responding only if the inputs it gets from some fixed random set of cells in the layer above form some definite pattern. … Each nerve cell fires and yields black output only if the inputs it gets from certain fixed positions match a particular template. The sequence of outputs from all the nerve cells can be used as a hash code, whose value tends to be the same for inputs that differ only by small changes.

The Significance of Universality in Rule 110 Practical computers and computer languages have traditionally been the only common examples of universality that we ever encounter. … For if one knew only about practical computers and about systems like the universal cellular automaton discussed early in this chapter , then one would probably assume that universality would rarely if ever be seen outside of systems that were specifically constructed to exhibit it. … So what this means is that if one looks at a sequence of systems with progressively more complicated rules, one should expect that the overall behavior they produce will become more complex only until the threshold of universality is reached.

The number of elements does end up increasing in this particular example, but only by a fixed amount at each step. … And as it turns out, among substitution systems with the same type of rules, all those which yield slow growth also seem to produce only such simple repetitive patterns. … The right-hand view shows that the rates of creation and destruction of elements are balanced closely enough that the total number of elements grows by only a fixed amount at each step.

But from what we have seen in this section such behavior appears to be quite rare: unlike many of the simple rules that we have discussed in this book, it seems that almost all simple constraints lead only to fairly simple patterns. … The constraint specifies that only the 56 3×3 templates shown at left can occur anywhere in the pattern, with the first template appearing at least once.

Discreteness in computer programs The reason for discreteness in computer programs is that the only real way we know how to construct such programs is using discrete logical structures. … And it is then certainly more consistent to make both data and programs involve only discrete elements. In Chapter 12 I will argue that this approach is not only convenient, but also necessary if we are to represent our computations using processes that can actually occur in nature.

The pattern produced continues to expand on both left and right, but only the part that fits across the page is shown here.

The first stage, illustrated in the top picture on the next page , is to get a cyclic tag system to emulate an ordinary tag system with the property that its rules depend only on the very first element that appears at each step. … Unlike the examples of symbolic systems in Chapter 3 , which involve only one symbol, these symbolic systems involve three symbols, p , q and r .