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those steps at which one of the two registers has just decreased to zero. And in this picture one immediately sees some apparently random variation in the instructions that are executed.

Part (c) of the picture then shows which instructions are executed for the first 400 times one of the registers has just decreased to zero. And part (d) finally shows the base 2 digits of the successive values attained by the second register when the first register has just decreased to zero. The results appear to show considerable randomness.

So even though it may not be as obvious as in some of the other systems we have studied, the simple register machine on the facing page can still generate complex and seemingly quite random behavior.

So what about more complicated register machines?

An obvious possibility is to allow more than two registers. But it turns out that very little is normally gained by doing this. With three registers, for example, seemingly random behavior can be obtained with a program that is seven rather than eight instructions long. But the actual behavior of the program is almost indistinguishable from what we have already seen with two registers.

Another way to set up more complicated register machines is to extend the kinds of underlying instructions one allows. One can for example introduce instructions that refer to two registers at a time, adding, subtracting or comparing their contents. But it turns out that the presence of instructions like these rarely seems to have much effect on either the form of complex behavior that can occur, or how common it is.

Yet particularly when such extended instruction sets are used, register machines can provide fairly accurate idealizations of the low-level operations of real computers. And as a result, programs for register machines are often very much like programs written in actual low-level computer languages such as C, Basic, Java or assembler.

In a typical case, each variable in such a program simply corresponds to one of the registers in the register machine, with no arrays or pointers being allowed. And with this correspondence, our general results on register machines can also be expected to apply to simple programs written in actual low-level computer languages.

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