Get Ackermann Function essential facts below. View Videos or join the Ackermann Function discussion. Add Ackermann Function to your PopFlock.com topic list for future reference or share this resource on social media.
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a totalcomputable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann-Péter function, is defined as follows for nonnegative integers m and n:
Its value grows rapidly, even for small inputs. For example, is an integer of 19,729 decimal digits (equivalent to 265536−3, or 22222−3).
and for p > 2 it extends these basic operations in a way that can be compared to the hyperoperations:
(Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann's function that are specifically designed for that purpose--such as Goodstein'shyperoperation sequence.)
In On the Infinite,David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert's personal secretary and former student, who actually proved the hypothesis in his paper On Hilbert's Construction of the Real Numbers.
Compared to most other versions Buck's function has no unessential offsets:
Definition and properties
Ackermann's original three-argument function is defined recursively as follows for nonnegative integers m, n, and p:
Of the various two-argument versions, the one developed by Péter and Robinson (called "the" Ackermann function by some authors) is defined for nonnegative integers m and n as follows:
It may not be immediately obvious that the evaluation of always terminates. However, the recursion is bounded because in each recursive application either m decreases, or m remains the same and n decreases. Each time that n reaches zero, m decreases, so m eventually reaches zero as well. (Expressed more technically, in each case the pair (m, n) decreases in the lexicographic order on pairs, which is a well-ordering, just like the ordering of single non-negative integers; this means one cannot go down in the ordering infinitely many times in succession.) However, when m decreases there is no upper bound on how much n can increase--and it will often increase greatly.
The Péter-Ackermann function can also be expressed in relation to various other versions of the Ackermann function:
( and would correspond with and , which could logically be added.)
For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n (at most exponentially). For , however, it grows much more quickly; even is about 2×1019728, and the decimal expansion of is very large by any typical measure.
An interesting aspect of the (Péter-)Ackermann function is that the only arithmetic operation it ever uses is addition of 1. Its fast growing power is based solely on nested recursion. This also implies that its running time is at least proportional to its output, and so is also extremely huge. In actuality, for most cases the running time is far larger than the output; see below.
A single-argument version that increases both m and n at the same time dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi- and superfactorial functions, and even functions defined using Knuth's up-arrow notation (except when the indexed up-arrow is used). It can be seen that f(n) is roughly comparable to f?(n) in the fast-growing hierarchy. This extreme growth can be exploited to show that f, which is obviously computable on a machine with infinite memory such as a Turing machine and so is a computable function, grows faster than any primitive recursive function and is therefore not primitive recursive.
In a category with exponentials, using the isomorphism (in computer science, this is called currying), the Ackermann function may be defined via primitive recursion over higher-order functionals as follows:
The function defined in this way agrees with the Ackermann function defined above: .
Number of recursions before the return of Ackerman(3,3)
To see how the Ackermann function grows so quickly, it helps to expand out some simple expressions using the rules in the original definition. For example, one can fully evaluate in the following way:
To demonstrate how 's computation results in many steps and in a large number:
Table of values
Computing the Ackermann function can be restated in terms of an infinite table. First, place the natural numbers along the top row. To determine a number in the table, take the number immediately to the left. Then use that number to look up the required number in the column given by that number and one row up. If there is no number to its left, simply look at the column headed "1" in the previous row. Here is a small upper-left portion of the table:
Values of A(m, n)
265536 - 3
The numbers here which are only expressed with recursive exponentiation or Knuth arrows are very large and would take up too much space to notate in plain decimal digits.
Despite the large values occurring in this early section of the table, some even larger numbers have been defined, such as Graham's number, which cannot be written with any small number of Knuth arrows. This number is constructed with a technique similar to applying the Ackermann function to itself recursively.
This is a repeat of the above table, but with the values replaced by the relevant expression from the function definition to show the pattern clearly:
Values of A(m, n)
0 + 1
1 + 1
2 + 1
3 + 1
4 + 1
n + 1
A(0, A(1, 0)) = A(0, 2)
A(0, A(1, 1)) = A(0, 3)
A(0, A(1, 2)) = A(0, 4)
A(0, A(1, 3)) = A(0, 5)
A(0, A(1, n-1))
A(1, A(2, 0)) = A(1, 3)
A(1, A(2, 1)) = A(1, 5)
A(1, A(2, 2)) = A(1, 7)
A(1, A(2, 3)) = A(1, 9)
A(1, A(2, n-1))
A(2, A(3, 0)) = A(2, 5)
A(2, A(3, 1)) = A(2, 13)
A(2, A(3, 2)) = A(2, 29)
A(2, A(3, 3)) = A(2, 61)
A(2, A(3, n-1))
A(3, A(4, 0)) = A(3, 13)
A(3, A(4, 1)) = A(3, 65533)
A(3, A(4, 2))
A(3, A(4, 3))
A(3, A(4, n-1))
A(4, A(5, 0))
A(4, A(5, 1))
A(4, A(5, 2))
A(4, A(5, 3))
A(4, A(5, n-1))
A(5, A(6, 0))
A(5, A(6, 1))
A(5, A(6, 2))
A(5, A(6, 3))
A(5, A(6, n-1))
Proof that the Ackermann function is not primitive recursive
Specifically, one shows that to every primitive recursive function there exists a non-negative integer such that for all non-negative integers ,
Once this is established, it follows that itself is not primitive recursive, since otherwise putting would lead to the contradiction .
The proof proceeds as follows: define the class of all functions that grow slower than the Ackermann function
and show that contains all primitive recursive functions. The latter is achieved by showing that contains the constant functions, the successor function, the projection functions and that it is closed under the operations of function composition and primitive recursion.
Since the function considered above grows very rapidly, its inverse function, f-1, grows very slowly. This inverse Ackermann functionf-1 is usually denoted by ?. In fact, ?(n) is less than 5 for any practical input size n, since is on the order of .
This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure and Chazelle's algorithm for minimum spanning trees. Sometimes Ackermann's original function or other variations are used in these settings, but they all grow at similarly high rates. In particular, some modified functions simplify the expression by eliminating the -3 and similar terms.
A two-parameter variation of the inverse Ackermann function can be defined as follows, where is the floor function:
This function arises in more precise analyses of the algorithms mentioned above, and gives a more refined time bound. In the disjoint-set data structure, m represents the number of operations while n represents the number of elements; in the minimum spanning tree algorithm, m represents the number of edges while n represents the number of vertices.
Several slightly different definitions of exist; for example, is sometimes replaced by n, and the floor function is sometimes replaced by a ceiling.
Other studies might define an inverse function of one where m is set to a constant, such that the inverse applies to a particular row. 
The inverse of the Ackermann function is primitive recursive.
Use as benchmark
The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion. The first published use of Ackermann's function in this way was in 1970 by Drago? Vaida and, almost simultaneously, in 1971, by Yngve Sundblad.
Sundblad's seminal paper was taken up by Brian Wichmann (co-author of the Whetstone benchmark) in a trilogy of papers written between 1975 and 1982.
Pettie, S. (2002). "An inverse-Ackermann style lower bound for the online minimum spanning tree verification problem". The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings: 155-163. doi:10.1109/SFCS.2002.1181892.
Sundblad, Yngve (March 1971). "The Ackermann function. A theoretical, computational, and formula manipulative study". BIT Numerical Mathematics. 11 (1): 107-119. doi:10.1007/BF01935330.
Vaida, Drago? (1970). "Compiler Validation for an Algol-like Language". Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série. 14 (60) (4): 487-502. JSTOR43679758.
Wichmann, Brian A. (March 1976). "Ackermann's function: A study in the efficiency of calling procedures". BIT Numerical Mathematics. 16: 103-110. doi:10.1007/BF01940783.
Wichmann, Brian A. (July 1977). "How to call procedures, or second thoughts on Ackermann's function". BIT Numerical Mathematics. 16: 103-110. doi:10.1002/spe.4380070303.