Has anything useful come from Ackermann's Function?

Question

Here is the function:

if (m == 0)
    return n + 1;
else if (n == 0)
    return A(m-1, 1);
else
    return A(m-1, A(m, n-1));

This seems like an interesting function, especially since its values grow quite rapidly (my computer crashes if I try to run A(4,2) or A(3,10)).

From the wikipedia page it seems like it was only invented to show that a total computable function does not have to be primitive recursive. Has anything practical come from this function, other than homework problems in a computer science class?

Answer 1

The inverse of Ackermann's function is useful in computational complexity, for instance in line arrangements and other problems in computational geometry. It's appearance in the analysis of some concrete problems and algorithms is somewhat surprising. See

• Davenport–Schinzel sequences and their role in lower envelopes
• Davenport-Schinzel Sequences and Their Geometric Applications

Answer 2

We used the inverse of Ackermann's function in section 5 of our paper “Untangling planar graphs from a specified vertex position - Hard cases” devoted to straight line drawings of planar graphs.