Chapter 3: The World of Simple Programs

Section 4: Turing Machines

Distribution of behavior [in Turing machines]

With 2 possible states and 2 possible colors for each cell, starting from a blank tape, the maximum repetition period obtained is 9 steps, and 12 out of the 4096 possible rules (or about 0.29%) yield non-repetitive behavior. With 3 states and 2 colors, the maximum period is 24, and about 0.37% of rules yield non-repetitive behavior, always nested. (Usually I have not found more complicated behavior in such rules even with initial conditions in which there are both black and white cells, though see page 761.) With 2 states and 3 colors, the maximum repetition period is again 24, about 0.65% of rules yield non-repetitive behavior, and the 14 rules discussed on page 709 yield more complex behavior. With more colors or more states, the percentage of rules that yield non-repetitive behavior steadily increases, as shown below, roughly like 0.28 (s - 1) (k - 1). (Compare page 1120.)

Image Source Notebooks:

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