And scientifically it is also unsatisfying to have to say that there are features of our universe which are not determined by any finite set of underlying rules, but are instead in a sense just pure accidents of history associated with the particular path that we have happened to follow in a multiway system.

But as we have seen a great many times in the course of the book, what emerges from a program can be very different from what is obvious in its underlying rules.

But as a first approximation, one can consider A collision between localized structures in the rule 110 class 4 cellular automaton.

Just like other sequences discussed in this chapter they are in some sense not, since they can both be generated by simple underlying rules.

Much like a multiway system, a non-deterministic Turing machine has rules that allow multiple choices to be made at each step, leading to multiple possible paths of evolution.

For even if one takes these processes to be pure quantum ones, what I believe is that in almost all cases appropriate idealized limits of them will reproduce what are in effect the usual rules for observations in quantum theory. … For the basic rules for pure quantum processes are entirely reversible (unitary). … I suspect, however, that in fact the most important source of randomness in most cases will instead be the phenomenon of intrinsic randomness generation that I first discovered in systems like the rule 30 cellular automaton.

And somehow whatever rules update the network must preserve this feature.

But the crucial idea that underlies the traditional approach to mathematical proof is that one should also be able to deduce such results just by manipulating expressions in purely symbolic form, using the rules of an axiom system, without ever having to do anything like filling in explicit values of variables.

The vast majority of forms created by humans in the past—say in art or architecture—have had basic features that are either directly copied from systems in nature, or are in effect built up by using extremely simple kinds of rules.

History of branching models The concept of systematic rules for the way that stems—particularly those carrying flowers—are connected in a plant seems to have been clearly understood among botanists by the 1800s.