Human generation of randomness If asked to type a random sequence of 0's and 1's, most people will at first produce a sequence with too many alternations between 0 and 1. But with modest learning time my experience is that one can generate sequences with quite good randomness.

Information content of initial conditions Common sense suggests that it is a quite different thing to specify a simple initial condition containing, say, a single black cell on a white background, than to specify an initial condition containing an infinite sequence of randomly chosen cells. … And as a result, mathematicians may find it difficult to understand my distinction between randomness generated intrinsically by the evolution of a system and randomness from initial conditions (see page 299 ). … For such systems cannot meaningfully be given infinite random initial conditions, yet they can still perfectly well generate highly random behavior.

But in fact, as discussed in Chapter 7 , randomness and complexity tend to lead to more, rather than less, robustness in overall behavior. Indeed, many even seemingly simple biological processes appear to be stabilized by randomness—leading, for example, to random fluctuations in interbeat intervals for healthy hearts. And some biological processes rely directly on complex or random phenomena—for example, finding good paths for foraging for food, avoiding predators or mounting suitable immune responses.

Random initial conditions [for squaring and primes CAs] The pictures below show the squaring and primes cellular automata starting from random initial conditions. … Changing these cases can lead to different behavior with random initial conditions.

Origins of randomness [in animal patterning] The model in the main text assumes that randomness enters through initial conditions. … But usually even the two sides of a single animal are never physically together, and they normally end up having quite uncorrelated random features. In cases such as fingerprints and zebra stripes there is some correlation between different sides, suggesting an intrinsic component to the randomness that occurs.

Sometimes the behavior is purely repetitive, but often it has features that seem effectively random. … The background simply repeats every 4 steps, but the main pattern has a complex and in many respects random form.

For while The behavior of a sequence of cellular automaton programs obtained by successive random mutations. … Each program in the sequence differs from the previous one by a single mutation, made completely at random.

[No text on this page] Examples of reversible cellular automata starting from random and from simple initial conditions.

Instead of exhibiting progressively more random behavior, it appears to fluctuate between quite ordered and quite disordered states.

Origin of probabilities Probabilities are normally assumed to enter for at least two reasons: (a) because of random variation between individuals, and (b) because of random errors in measurement. … Probabilistic models for abstract mathematical systems have in the past been rare, though the results about randomness in this book may make them more common in the future.