Defining the Notion of Randomness…So how then can we develop a useful yet precise definition of randomness? … So this suggests a definition that can be given of randomness: something should be considered to be random whenever there is essentially no simple program that can succeed in detecting regularities in it. … So this means that the everyday definition of randomness that we discussed at the very beginning of this section is in the end already quite unambiguous.

Randomness from the Environment…where the sparks occur depends on the detailed microscopic motion of the molecules in the gas, and is therefore potentially quite random. … For one might think that microscopic physical processes would always produce the best possible randomness. But there are two important effects which tend to limit this randomness, or indeed any randomness that is obtained through the mechanism of interaction with the environment.

The Intrinsic Generation of Randomness…[No text on this page] Examples of three so-called linear congruential random number generators. … Multiplier 65539 was used as the random number generator on many computer systems, starting with mainframes in the 1960s. … If the output from the generator was perfectly random, then in each case these points would be uniformly distributed.

The Intrinsic Generation of Randomness…At first, such failures might suggest that more complicated schemes must be needed if one is to get good randomness. … But although some aspects of the behavior of such systems can be made quite random, deviations from perfect randomness are still often found. And seeing this one might conclude that it must be essentially impossible to produce good randomness with any kind of system that has reasonably simple rules.

Randomness from the Environment…What one finds is that the grain jumps around in an apparently random way. … But to observe random Brownian motion, one needs a microscope. … But in fact such randomness is quite obvious in the operation of many kinds of electronic devices.

Note (g) for Randomness in Class 3 Systems…Rule 94 [with simple initial conditions] With appropriate initial conditions this class 2 rule can yield both nested and random behavior, as shown below.

The Intrinsic Generation of Randomness…[No text on this page] The effects of various levels of external randomness on the behavior of continuous cellular automata with generalizations of rules 90 and 30. … In both cases, every value at each step is also perturbed by a random amount up to the percentage indicated for each picture.

Defining the Notion of Randomness…For it can be generated just by Pictures exhibiting different degrees of apparent randomness. Pictures (a) and (b) have obvious regularities, and would never be considered particularly random. But picture (c) has almost no obvious regularities, and would typically be considered quite random.

Note (a) for Randomness in Class 3 Systems…The origin of this phenomenon is that with completely random initial conditions rule 73 evolves to a collection of independent regions, as in the picture below, and many of these regions contain patterns that repeat with period 3.

Note (f) for Randomness in Class 3 Systems…With a background consisting of repetitions of the block , insertion of a single initial white cell yields a largely random pattern that expands by one cell per step.