One might have thought that in the literature of traditional science new models would be proposed all the time. … And among the models that have been used, almost all those that have gone beyond the level of being purely descriptive have ended up being formulated in very much the same kind of way: typically as collections of mathematical equations.

For all it takes is that systems in nature operate like typical programs and then it follows that their behavior will often be complex. … One might have thought that with all their successes over the past few centuries the existing sciences would long ago have managed to address the issue of complexity.

The pictures immediately remind one of the overall branching patterns of all sorts of plants—from algae to ferns to trees to many kinds of flowering plants. … It has never in the past been at all clear how leaves get the shapes they do.

In general, however, the maximum possible period for a system containing a certain number of cells can be achieved only if the evolution of the system from any initial condition eventually visits all the possible states of the system, as discussed on page 258 . … And instead, starting from any particular initial condition, the system will only ever visit a tiny fraction of all possible states.

And indeed, as the picture below shows, it is not difficult to construct multiway systems in which all possible strings of a particular kind are produced. … And indeed, in any multiway system with a limited set of rules, such sequences must necessarily be subject to all sorts of constraints.

And in fact it is my strong suspicion that the kinds of programs that I have discussed in the past few sections will actually in the end turn out to show many if not all the key features of quantum theory. … But there are all sorts of phenomena in quantum theory that seem to indicate that electrons do not in fact behave like ordinary objects that have definite properties independent of us making observations of them.

there is causal invariance, then ultimately all these different histories must in a sense be equivalent. … But in fact—just as in so many other seemingly irreversible processes—all that is needed to preserve reversibility is that if one looks at sufficient details of the system there can be arbitrary and seemingly random behavior.

For in the course of the chapter we have discussed a whole range of different kinds of perception and analysis, yet in essentially all cases we have found that the overall capabilities they exhibit are rather similar. … Indeed, as one example, one could imagine just enumerating all possible simple descriptions of some particular type, and then testing in each case to see whether what one gets matches a piece of data that one has.

The first stripe carries the color of the left-hand neighbor, and causes all cases in the rule where that neighbor does not have the appropriate color to be eliminated. … And after all three stripes have passed, only one of the 8 cases ever survives, and this case is then the one that gives the new color for the cell.

And indeed in setting up a correspondence with rule 110, it is convenient to left-right reverse all pictures of cyclic tag systems. … Every line in the diagram corresponds to a single localized structure in rule 110, and although the whole diagram cannot be drawn completely to scale, the collisions between lines correctly show all the basic interactions that occur between structures.