Ackermann's function


Definition: A function of two parameters whose value grows very fast.

Formal Definition:

See also inverse Ackermann function.

Note: Many people have defined other similar functions which are not simply a restating of this one.

In 1928, Wilhelm Ackermann observed that A(x,y,z), the z-fold iterated exponentiation of x with y, is a recursive function that is not primitive recursive. A(x,y,z) was simplified to a function of 2 variables by Rózsa Péter in 1935. Raphael M. Robinson simplified the initial condition in 1948.

Author: PEB


History of the function and (Modula-2) code.

More information

Robert Munafo's Versions of Ackermann's Function and analysis. Cowles and Bailey Several Versions of Ackermann's function.

Wilhelm Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Mathematische Annalen 99(1):118-133, December 1928.

The formal definition given here is Gnx in the first page of
Raphael M. Robinson, Recursion and Double Recursion, Bulletin of the American Mathematical Society 54:987-993, October 1948.

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 18 November 2019.
HTML page formatted Mon Nov 18 11:40:56 2019.

Cite this as:
Paul E. Black, "Ackermann's function", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 18 November 2019. (accessed TODAY) Available from: