Looking just at the ordinary evolution labelled (a), however, the system might still appear to have quite simple and regular behavior. But a closer examination turns out to reveal irregularities. Part (b) of the picture shows a version of the evolution compressed to include only
A register machine whose behavior seems in some ways random. The program for this register machine is eight instructions long. Testing all 11,019,960,576 possible programs of length eight revealed just this and 125 similar cases of complex behavior. Part (b) shows the evolution in compressed form, with only those steps included at which either of the registers has just decreased to zero. The values of the nonzero registers are shown using a logarithmic scale. Part (c) shows the instructions that are executed for the first 400 times that one of the registers is decreased to zero. Finally, part (d) gives the successive values attained by the second register at steps where the first register has just decreased to zero. These values are given here as binary digit sequences. As discussed on page 122, the values can in fact be obtained by a simple arithmetic rule, without explicitly following each step in the evolution of the register machine. If one value is n, then the next value is 3n/2 if n is even, and (3n+1)/2 if n is odd. The initial condition is n=1.