Manuals/calci/ACKERMANN

ACKERMANN(m,n)

• $m$ and $n$ are the positive integers.

Description

• The Ackermann function is a classic example of a recursive function, notable especially because it is not a primitive recursive function.
• All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.
• Its arguments are never negative and it always terminates.
• One common version, the two-argument Ackermann–Péter function, is defined as follows:

$A(m,n)={\begin{cases}n+1{\mbox{if}}m=0\\A(m-1,1)&{\mbox{if}}m>0andn=0\\A(m-1,A(m,n-1))&{\mbox{if}}m>0andn>0\end{cases}}$ for nonnegative integers m and n.

• Its value grows rapidly, even for small inputs.