In the first case shown, starting for example at position 4 the dot then visits positions 5, 0, 1, 2 and so on, at each step going from one node in the network to the next. … There are cycles which contain states that are visited repeatedly, and there can also be trees that represent transient states that can each only ever occur at most once in the evolution of the system.

For I no longer control the basic rules of the systems I am studying, and instead I must just try to deduce these rules from observation—with the potential that despite my best efforts my deductions could simply be incorrect.

For some of the best established features of physical systems—such as conservation of energy or equivalence of directions in space—seem to have no obvious analogs in most of the programs we have discussed so far in this book.

so long as the structure of the network is kept the same, it is fairly easy even in this case to deduce from a given set of data what probabilities in the network provide the best model for the data—for essentially all one need do is to follow the path corresponding to the data, and see with what frequency each connection from each node ends up being used.

Card shuffling (see page 974 ) has been used as a source of randomness since at least the 1300s.

Two of the most important strategies are maintaining a whole population of individuals, not just the single best result so far, and using sex to produce large-scale mixing.

This does not seem to happen for rules that involve 4 neighbors, but with 8 neighbors there are cases in which clusters can get fairly large, but end up having no sites where further cells can be added.

For even in the very best case any block of cells in the input can never be compressed to less than one cell in the output.

No doubt in many cases the number of nodes that have to be visited eventually increases like 2 t , but many branches usually die off quickly, greatly reducing the typical effort required in practice.

And in the next few sections, I will discuss various kinds of systems where the randomness that is seen can be best described by each of the three mechanisms for randomness identified here.