Randomness in Class 3 Systems…Randomness in Class 3 Systems When one looks at class 3 systems the most obvious feature of their behavior is its apparent randomness. But where does this randomness ultimately come from? … But the crucial point that I discovered only some time later is that random behavior can also occur even when there is no randomness in initial conditions.

Randomness in Class 3 Systems…But the crucial point is that even without any initial randomness much of what we see in the second picture still looks like typical random class 3 behavior. … Do such systems always produce randomness even with simple initial conditions? … With random initial conditions typical class 3 behavior is seen.

Randomness in Class 3 Systems…And indeed when random initial conditions are used, rule 90 does manage to produce random behavior of the kind expected in class 3. … Examples of evolution from random initial conditions with a low density of black cells. … The reason for this difference is that in rule 22 the randomness we see is intrinsically generated by the evolution of the system, while in rule 90 it comes from randomness in the initial conditions.

Randomness in Class 3 Systems…There are however a few cellular automata in which class 3 behavior is obtained with random initial conditions, but in which no significant randomness is ever produced with simple initial conditions. … The only way to get a random pattern therefore is to have an infinite number of randomly placed black cells in the initial conditions. … The randomness in the last picture shown here is thus purely a consequence of the randomness in its initial conditions.

Randomness in Class 3 Systems…But in the bottom case, it is instead in many respects random, much like rule 30.

The Intrinsic Generation of Randomness…But it is also possible to get long random sequences by continual interaction with a random external environment, as in the first mechanism for randomness discussed in this chapter . … And the point is that in most cases, intrinsic randomness generation can produce similar randomness in a much shorter time. … The basic reason is that intrinsic randomness generation in a sense puts all the components in a system to work in producing new randomness, while getting randomness from the environment does not.

Three Mechanisms for Randomness…Three Mechanisms for Randomness In nature one of the single most common things one sees is apparent randomness. … And it could be that in each case the cause of randomness is different. … In the first mechanism, randomness is explicitly introduced into the underlying rules for the system, so that a random color is chosen for every cell at each step.

The Intrinsic Generation of Randomness…The Intrinsic Generation of Randomness In the past two sections [ 3 , 4 ], we have studied two possible mechanisms that can lead to observed randomness. … One of the important results of this book, however, is that there is also a third possible mechanism for randomness, in which no random input from outside is needed, and in which randomness is instead generated intrinsically inside the systems one is looking at. … There is no random input to this system, yet its behavior seems in many respects random.

Randomness from the Environment…Randomness from the Environment With the first mechanism for randomness discussed in the previous section , the randomness of any particular system is taken to be the result of continual interaction between that system and randomness in its environment. … There is nothing intrinsically random about the boat itself. … But what is the real origin of this apparent randomness?

Three Mechanisms for Randomness…Insofar as the initial conditions are random, therefore, so also will the sequence of colors of any particular cell be correspondingly random. … But the basic idea of this mechanism for randomness is that the randomness one sees arises from some kind of transcription of randomness that is present in the initial conditions. … The crucial point that we first saw on page 27 is that simple programs can produce apparently random behavior even when they are given no random input whatsoever.