So even if one allows rather general structure, the evidence is that in the end there is no way to set up any simple formula that will describe the outcome of evolution for a system like rule 30. … So what this means is that just like for every other method of analysis that we have considered, we have little choice but to conclude that traditional mathematics and mathematical formulas cannot in the end realistically be expected to tell us very much about patterns generated by systems like rule 30. … For in the course of this book we have seen a great many systems whose underlying rules are extremely simple, yet whose overall behavior is sufficiently complex that even by thinking quite hard we cannot recognize its simple origins.

So on the basis of traditional intuition, one might then assume that the way to solve this problem must be to use systems with more complicated underlying rules, perhaps more closely based on details of human psychology or neurophysiology. But from the discoveries in this book we know that this is not the case, and that in fact very simple rules are quite sufficient to produce highly complex behavior. … And it is in the end my strong suspicion that most of the core processes needed for general human-like thinking will be able to be implemented with rather simple rules.

When one learns a language—at least as a young child—one implicitly tends to deduce simple grammatical rules that are in effect specific generalizations of examples one has encountered. … Actual human languages normally have many exceptions to any simple grammatical rules. … But the fact that most modern computer languages are specifically set up to follow simple grammatical rules seems to make their structures particularly easy for us to learn—perhaps because they fit in well with low-level processes of human thinking.

But this is essentially like saying that once one knows the rules for a system nothing else about it should ever be considered interesting. 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. And the point is that if computational irreducibility is present, then there is in a sense all sorts of information about the behavior of a system that can only be found from its rules by doing an irreducibly large amount of computational work.

In many respects, the simpler the rules, the more likely it might seem that they could be associated with ordinary physical processes, without anything like intelligence being involved. Yet as we discussed above, if one could actually determine that the rules used in a given case were the simplest possible, then this might suggest that they were somehow set up on purpose. But in practice if one just receives a signal one normally has no way to tell which of all possible rules for producing it were in fact used.

And indeed my guess is that the essential features of all sorts of intricate structures that are seen in living systems can actually be reproduced with remarkably simple rules—making it for example possible to use technology to repair or replace a whole new range of functions of biological tissues and organs. But given some form of perhaps complex behavior, how can one find rules that will manage to generate it? 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.

Sequential substitution systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a sequential substitution system which emulates it: CAToSSS[rules_] := Join[rules /.

Rule expressions [for cellular automata] The table below gives Boolean expressions for each of the elementary rules.

Rule 45 The center column of the pattern appears for practical purposes random, just as in rule 30.

[Cellular automaton] rule emulations The network below shows which quiescent symmetric elementary rules can emulate which with blocks of length 8 or less. … In all cases things are set up so that several steps in one rule emulate a single step in another. … So this means that one can consider encodings based on blocks that have a kind of staircase shape—as in the rule 45 example shown.