Chapter 6: Starting from Randomness

Section 2: Four Classes of Behavior

Random initial conditions in other systems

Whenever the initial conditions for a system can involve an infinite sequence of elements these elements can potentially be chosen at random. In systems like mobile automata and Turing machines the colors of initial cells can be random, but the active cell must start at a definite location, and depending on the behavior only a limited region of initial cells near this location may ever be sampled. Ordinary substitution systems can operate on infinite sequences of elements chosen at random. Sequential substitution systems, however, rely on scanning limited sequences of elements, and so cannot readily be given infinite random initial conditions. The same is true of ordinary and cyclic tag systems. Systems based on continuous numbers involve infinite sequences of digits which can readily be chosen at random (see page 154). But systems based on integers (including register machines) always deal with finite sequences of digits, for which there is no unique definition of randomness. (See however the discussion of number representations on page 1070.) Random networks (see pages 963 and 1038) can be used to provide random initial conditions for network systems. Multiway systems cannot meaningfully be given infinite random initial conditions since these would typically lead to an infinite number of possible states. Systems based on constraints do not have initial conditions. (See also page 920.)

From Stephen Wolfram: A New Kind of Science [citation]