Ackermann Function

+The **Ackermann function**, often denoted as $A(m, n)$, is a foundational example in [computability](/wiki/computability), famed for its exceptionally rapid growth. Defined through nested [recursion](/wiki/recursion) for two non-negative integers $m$ and $n$, it is one of the simplest and earliest examples of a [total computable function](/wiki/computable_function) that is not a [primitive recursive function](/wiki/primitive_recursion). This property makes it a significant concept in [theoretical computer science](/wiki/computer_science) and [mathematical logic](/wiki/logic).

+Its growth rate is so extreme that even small input values quickly produce unimaginably large numbers, making it impractical for direct computation with anything but very small inputs. For instance, $A(4, 2)$ is already a number with many thousands of digits. The function demonstrates the distinction between general [recursive functions](/wiki/recursion) and primitive recursive functions, highlighting the limits of computational expressiveness for simpler recursive schemes. It also finds applications in the analysis of [algorithms](/wiki/algorithms), particularly in the study of [union-find data structures](/wiki/union_find) where a related [inverse Ackermann function](/wiki/inverse_ackermann) appears, offering a very slowly growing upper bound for the amortized time complexity.

+The **Ackermann function** is a foundational example in [computability](/wiki/computability), famed for its exceptionally rapid growth. Defined through nested [recursion](/wiki/recursion), it transcends the class of primitive recursive functions, showcasing the bounds of simple definable computation.

+## See also

+- [Recursive Function](/wiki/recursive_function)

+- [Growth Rates](/wiki/growth_rates)

+- [Computational Complexity](/wiki/computational_complexity)