The Intrinsic Generation of Randomness…But just how random is this sequence really? … But is it truly random? … And according to most of these definitions, the sequence is indeed truly random.

The Intrinsic Generation of Randomness…So why do linear congruential generators not produce better randomness? … The basic answer, I believe, is that whenever a large amount of randomness is produced in a short time, intrinsic randomness generation is overwhelmingly likely to be the mechanism responsible. We saw in the previous section that random details of the initial conditions for a system can lead to a certain amount of randomness in

The Intrinsic Generation of Randomness…As we first saw on page 119 , the patterns of digits obtained in this way seem quite random. And the idea of so-called linear congruential random number generators is precisely to make use of this randomness. … Yet even with this restriction, the sequences generated are random enough that at least until recently they were almost universally what was used as a source of randomness in practical computing.

The Intrinsic Generation of Randomness…And what this means is that when intrinsic randomness generation is the dominant mechanism it is indeed realistic to expect at least some level of repeatability in the random behavior one sees in real experiments. … So in all of these cases the randomness one sees cannot reasonably be attributed to randomness that is introduced from the environment—either continually or through initial conditions. And instead, there is no choice but to conclude that the randomness must in fact come from the mechanism of intrinsic randomness generation that I have discovered in simple programs, and discussed in this section .

Defining the Notion of Randomness…So one possibility is to define randomness so that something is considered random only if no short description whatsoever exists of it. … So to say that something is random only if no short description whatsoever exists of it turns out to be a highly restrictive definition of randomness. And in fact, as I mentioned in Chapter 7 , it essentially implies that no process based on definite rules can ever manage to generate randomness when there is no randomness before.

Defining the Notion of Randomness…Defining the Notion of Randomness Many times in this book I have said that the behavior of some system or another seems random. But so far I have given no precise definition of what I mean by randomness. … So given this everyday notion of randomness, how can we build on it to develop more precise definitions?

Note (e) for Randomness in Class 3 Systems…The base 2 digit sequences for 19, 25, 37, 39, 41, 45, 47, 51, 57, 61, ... also give initial conditions that yield randomness. Despite its overall randomness there are some regularities in the pattern shown at the bottom of the page. The overall density of black cells is not 1/2 but is instead approximately 0.35, just as for random initial conditions.

The Intrinsic Generation of Randomness…And as a result, the center column of rule 30 cannot be considered truly random according to such definitions. … So whether or not one chooses to say that the sequence is truly random, it is, as far as one can tell, at least random for all practical purposes. And in fact sequences closely related to it have been used very successfully as sources of randomness in practical computing.

The Intrinsic Generation of Randomness…Yet having said this, one can ask how one can tell in an actual experiment on some particular system in nature to what extent intrinsic randomness generation is really the mechanism responsible for whatever seemingly random behavior one observed. … But the point is that with the mechanism of intrinsic randomness generation, there is no dependence on the environment. … And so for example the sequence of colors of the center cell, while seemingly random, will also be exactly the same.

Defining the Notion of Randomness…In fact, if one followed such a definition most of the pictures in this book that I have said look random—including for example picture (c) on page 553 —would be considered not random. And following the discussion of Chapter 7 , so would at least many of the phenomena in nature that we normally think of as random. … Over the years, a variety of definitions of randomness have been proposed that are based on the absence of certain specific regularities.