And if one is going to reproduce all the equivalences that hold for a particular form then these constraints must in effect be such as to force that form to occur. … And for each of these only a limited number of forms are allowed—all of which ultimately turn out to be equivalent to just the single forms shown on the facing page . … Axiom systems that can be used to derive all the equivalences between expressions that involve operators with the forms shown.

Yet most of this book is concerned precisely with all the interesting behavior that can emerge even if one knows the rules for a system. … So what this suggests is that a theorem might be considered interesting not only if it cannot be derived at all from simpler theorems but also if it cannot be derived from them except by some long proof. … In logic, however, all proofs are in effect ultimately of limited length.

At first, all of these might seem like reasonable indicators of true intelligence. … Adaptation to all sorts of complex situations also occurs in a great many systems. … So given all of this is there any way to define a general notion of true intelligence?

In the past biology—with all its details of DNA, proteins, ribosomes and so on—has provided our only example of programmable construction on an atomic scale. … The traditional engineering approach—if it works at all—will almost inevitably give rules that are in effect at least as complicated as the behavior one is trying to get. … But in my experience there are so many different and unexpected things that can happen with simple programs that ultimately the only way to find what one wants is essentially just to do an exhaustive search of all possibilities.

All is computation The early history of science includes many examples of attempts to treat all aspects of the universe in a uniform way. … "All is fire" was never definite enough to lead to much, but "all is number" can be viewed as an antecedent to the whole application of mathematics to science, and "all is atoms" to the atomic theory of matter and quantum mechanics. My "all is computation" will, I believe, form the basis for a fruitful new direction in science.

And in fact, all the systems that we have considered so far in this book can be thought of as having the same simple structure in time. For all of them are ultimately set up just to evolve progressively from one state to the next. … The main feature of multiway systems is that all the distinct sequences that result are kept.

Then the point is that because the rules depend only on the color of a particular branch, and not on the colors of any neighboring branches, the subtrees that are generated from all the branches of the same color must have exactly the same structure, as in the pictures below. … Each rule yields a different sequence of elements, but all of them ultimately have simple nested forms.

Extended versions exist of all but structures (a) and (j).

[No text on this page] Examples of network evolution according to the same basic underlying rules as on page 511 , but now with all possible clusters of nodes that do not overlap being replaced at each step.

Many of the maps I will consider can be expressed in terms of standard mathematical functions, but in general all that is needed is that the map take any possible number between 0 and 1 and yield some definite number that is also between 0 and 1. … On the first of the next two pages all the examples start with the number 1/2 —which has a simple digit sequence. … For as the picture illustrates, the so-called shift map used in case (d) simply serves to shift all digits one position to the left at each step.