So are there computations that take still longer to do? In Turing machine (f) the maximum number of steps increases exponentially with the length of the input. But unlike in example (e), this Turing machine is not the only one that computes the function it computes. And in fact both (g) and (h) compute the same function—but in a linearly increasing and constant number of steps respectively.

So what about other Turing machines? In general there is no guarantee that a particular Turing machine will ever even complete a computation in a finite number of steps. For as happens with several inputs in examples (i) and (j) the head may end up simply going further and further to the left—and never get to the point on the right that is needed for the computation to be considered complete.

But if one ignores inputs where this happens, then at least in examples (i) and (j) the maximum number of steps still grows in a very systematic linear way with the length of the input.

In example (k), however, there is more irregular growth. But once again the maximum number of steps in the end just increases like the square of the length of the input. And indeed if one looks at all 4096 Turing machines with 2 states and 2 colors it turns out that the only rates of growth that one ever sees are linear, square and exponential.

And of the six examples where exponential growth occurs, all of them are like example (f) above—so that there is another 2-state 2-color Turing machine that computes the same function, but without the maximum number of steps increasing at all with input length.

So what happens if one considers more complicated Turing machines? With 3 states and 2 colors there are a total of 2,985,984 possible machines. And it turns out that there are about 33,000 distinct functions that one or more of these machines computes.

Most of the time the fastest machine at computing a given function again exhibits linear or at most quadratic growth. But the facing page shows some cases where instead it exhibits exponential growth.

And indeed in a few cases the growth seems to be even faster. Example (h) is the most extreme among 3-state 2-color Turing machines: with the size 7 input 106 it already takes 1,978,213,883 steps

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