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The Ackermann function a(m, n) is a classical example of a total recursive function which is not primitive recursive. It grows faster than any primitive recursive function. It is usually defined by a general recurrence together with two “boundary” conditions. In this paper we obtain a closed form of a(m, n), which involves the Knuth superpower notation, namely a(m, n) = 2 m−2 ↑ (n + 3) − 3. Generalized Ackermann functions, that is functions satisfying only the general recurrence and one of the boundary conditions are also studied. In particular, we show that the function 2 m−2 ↑ (n + 2)− 2 also belongs to the “Ackermann class”.

Ackermann and the superpowers