One approach to getting around this suggested in the late 1950s is the many-worlds interpretation (see page 1035 ): that there is in a sense a universal pure quantum process that involves all possible outcomes for every conceivable observation, and that represents the tree of all possible threads of history—but that in a particular thread, involving a particular sequence of tree branches, and representing a particular thread of experience for us, there is in effect a reduction in the pure quantum process at each branch point. … 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. … But at least for a single particle, the Schrödinger equation is in all ways linear, and so it cannot support any kind of real sensitivity to initial conditions, or even to parameters.

Lower bounds [on computational complexity] If one could prove for example that P ≠ NP then one would immediately have lower bounds on all NP-complete problems. … An example is testing whether one can match all possible sequences with a regular expression that involves s -fold repetitions.

First of all, at a formal level, equations corresponding to these two cases can look very similar. And secondly, the equations are almost always so difficult to deal with at all that distinctions between the two cases are not readily noticed.

And while there are various triangulation methods that for example avoid triangles with small angles, no standard method yields networks analogous to the ones I consider in which all triangle edges are effectively the same length. … And indeed a crucial point for my discussion in the main text is that in formulating general relativity one actually does not appear to need all the structure of a simplicial complex. … And in general, some—but not all—of the standard constructions used in continuous spaces can also immediately be used in networks.

One might think that with all the mathematics developed for studying systems based on numbers it would be easy to answer these kinds of questions.

In the layout shown here, all the networks have their nodes arranged along a line.

But at least in all the examples below, the patterns that occur in each individual region are still simple and repetitive.

And in this case it turns out that all patterns are in effect just simple superpositions of the basic nested pattern that is obtained by starting with a single black cell.

And in fact, in terms of such digit sequences, the kneading process consists simply in shifting all digits one place to the left at each step, as shown in the pictures below.

A common approach used both in natural systems and in practical computing is to have some form of iterative procedure, in which one starts from a pattern chosen at The fraction of all possible patterns in which a certain percentage of squares violate the constraints discussed on page 211 .