Note (d) for Randomness from the Environment

Note (b) for Randomness from the Environment

Note (c) for The Intrinsic Generation of Randomness

Note the frequent occurrence of seemingly random turbulence.

Indeed, as the pictures on the next page demonstrate, such systems can produce considerable randomness even when starting from very simple initial conditions. … In rule (b), they move in fairly simple ways, and in rules (c) and (d), they move in a seemingly somewhat random way.

But as we saw in the previous chapter , it is also possible for cellular automata to produce patterns that seem in many respects random. And out of the 256 rules discussed here, it turns out that 10 yield such apparent randomness.

Indeed, in the particular case of systems such as random walks, the Central Limit Theorem suggested over two centuries ago ensures that for a very wide range of underlying microscopic rules, the same continuous so-called Gaussian distribution will always be obtained. … The pictures on the next page show, for example, what happens if one looks at two-dimensional random walks on square and hexagonal lattices. … A demonstration of the fact that for a wide range of underlying rules for each step in a random walk, the overall distribution obtained always has the same continuous form.

Most often what happens is that on a small scale a system exhibits randomness, but on a larger scale this randomness averages out to leave apparent uniformity, as in the fourth set of pictures below. … Averaging out small-scale randomness yields apparent uniformity, as shown here for a rule 30 pattern.

Randomized algorithms Whether a randomized algorithm gives correct answers can be viewed as a test of randomness for whatever supposedly random sequence is provided to it. … And this is basically why it has so often proved possible to replace randomized algorithms by deterministic ones that are at least as efficient (see page 1192 ). An example is Monte Carlo integration, where what ultimately matters is uniform sampling of the integrand—which can usually be achieved better by quasi-random irrational number multiple (see page 903 ) or digit reversal (see page 905 ) sequences than by sequences one might consider more random.

Applications of randomness Random drawing of lots has been used throughout recorded history as an unbiased way to distribute risks or rewards. … Such randomness must be repeatable. … In the past, randomness was usually viewed as a thing to be avoided.