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Ackermann function

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable 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 non-negative 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 is the two-argument Ackermann–Péter function developed by Rózsa Péter and Raphael Robinson. Its value grows very rapidly; for example, $$\operatorname{A}(4, 2)$$ results in $$2^{65536} - 3$$, an integer of 19,729 decimal digits.

History
In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited with discovering total computable functions (termed simply "recursive" in some references) that are not primitive recursive. Sudan published the lesser-known Sudan function, then shortly afterwards and independently, in 1928, Ackermann published his function $$\varphi$$ (from Greek, the letter phi). Ackermann's three-argument function, $$\varphi(m, n, p)$$, is defined such that for $$p=0,1,2$$, it reproduces the basic operations of addition, multiplication, and exponentiation as

 * $$\begin{align}

\varphi(m, n, 0) &= m+n \\ \varphi(m, n, 1) &= m\times n \\ \varphi(m, n, 2) &= m^n \end{align}$$

and for p > 2 it extends these basic operations in a way that can be compared to the hyperoperations:

 * $$\begin{align}

\varphi(m, n, 3) &= m[4](n+1) \\ \varphi(m, n, p) &\gtrapprox m[p+1](n+1) && \text{for } p > 3 \end{align}$$

(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's hyperoperation 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.

Rózsa Péter and Raphael Robinson later developed a two-variable version of the Ackermann function that became preferred by almost all authors.

The generalized hyperoperation sequence, e.g. $$G(m, a, b) = a[m]b$$, is a version of the Ackermann function as well.

In 1963 R.C. Buck based an intuitive two-variable variant $$\operatorname{F}$$ on the hyperoperation sequence:

 * $$\operatorname{F}(m,n) = 2[m]n.$$

Compared to most other versions, Buck's function has no unessential offsets:

 * $$\begin{align}

\operatorname{F}(0,n) &= 2[0]n = n + 1 \\ \operatorname{F}(1,n) &= 2[1]n = 2 + n \\ \operatorname{F}(2,n) &= 2[2]n = 2 \times n \\ \operatorname{F}(3,n) &= 2[3]n = 2^n \\ \operatorname{F}(4,n) &= 2[4]n = 2^{2^{2^{{}^{.^{.^{{}_.2}}}}}} \\ &\quad\vdots \end{align}$$

Many other versions of Ackermann function have been investigated.

Definition: as m-ary function
Ackermann's original three-argument function $$\varphi(m, n, p)$$ is defined recursively as follows for nonnegative integers $$m,n,$$ and $$p$$:

 * $$\begin{align}

\varphi(m, n, 0) &= m + n \\ \varphi(m, 0, 1) &= 0 \\ \varphi(m, 0, 2) &= 1 \\ \varphi(m, 0, p) &= m && \text{for } p > 2 \\ \varphi(m, n, p) &= \varphi(m, \varphi(m, n-1, p), p - 1) && \text{for } n, p > 0 \end{align}$$

Of the various two-argument versions, the one developed by Péter and Robinson (called "the" Ackermann function by most authors) is defined for nonnegative integers $$m$$ and $$n$$ as follows:

\begin{array}{lcl} \operatorname{A}(0, n) & = & n + 1 \\ \operatorname{A}(m+1, 0) & = & \operatorname{A}(m, 1) \\ \operatorname{A}(m+1, n+1) & = & \operatorname{A}(m, \operatorname{A}(m+1, n)) \end{array} $$

The Ackermann function has also been expressed in relation to the hyperoperation sequence:
 * $$A(m,n) = \begin{cases}

n+1 & m=0 \\ 2[m](n+3)-3 & m>0 \\ \end{cases}$$

 * or, written in Knuth's up-arrow notation (extended to integer indices $$\geq -2$$):
 * $$ = \begin{cases}

n+1 & m=0 \\ 2\uparrow^{m-2} (n+3) - 3 & m>0 \\ \end{cases}$$
 * or, equivalently, in terms of Buck's function F:
 * $$ = \begin{cases}

n+1 & m=0 \\ F(m,n+3) - 3 & m>0 \\ \end{cases}$$

Definition: as iterated 1-ary function
Define $$f^{n}$$ as the n-th iterate of $$f$$:
 * $$\begin{array}{rll}

f^{0}(x) & = & x \\ f^{n+1}(x) & = & f(f^{n}(x)) \end{array}$$

Iteration is the process of composing a function with itself a certain number of times. Function composition is an associative operation, so $$f(f^{n}(x)) = f^{n}(f(x))$$.

Conceiving the Ackermann function as a sequence of unary functions, one can set $$\operatorname{A}_{m}(n) = \operatorname{A}(m,n)$$.

The function then becomes a sequence $$\operatorname{A}_0, \operatorname{A}_1, \operatorname{A}_2, ...$$ of unary functions, defined from iteration:

\begin{array}{lcl} \operatorname{A}_{0}(n) & = & n+1 \\ \operatorname{A}_{m+1}(n) & = & \operatorname{A}_{m}^{n+1}(1) \\ \end{array} $$

Computation
The recursive definition of the Ackermann function can naturally be transposed to a term rewriting system (TRS).

TRS, based on 2-ary function
The definition of the  2-ary  Ackermann function leads to the obvious reduction rules

\begin{array}{lll} \text{(r1)} & A(0,n)      & \rightarrow & S(n) \\ \text{(r2)} & A(S(m),0)   & \rightarrow & A(m,S(0)) \\ \text{(r3)} & A(S(m),S(n)) & \rightarrow & A(m,A(S(m),n)) \end{array} $$

Example

Compute $$A(1,2) \rightarrow_{*} 4$$

The reduction sequence is

To compute $$\operatorname{A}(m, n)$$ one can use a stack, which initially contains the elements $$\langle m,n \rangle$$.

Then repeatedly the two top elements are replaced according to the rules

\begin{array}{lllllllll} \text{(r1)} & 0    &,& n     & \rightarrow & (n+1) \\ \text{(r2)} & (m+1) &,& 0    & \rightarrow & m &,& 1 \\ \text{(r3)} & (m+1) &,& (n+1) & \rightarrow & m &,& (m+1) &,& n \end{array} $$

Schematically, starting from $$\langle m,n \rangle$$:

WHILE stackLength <> 1 {   POP 2 elements; PUSH 1 or 2 or 3 elements, applying the rules r1, r2, r3 }

The pseudocode is published in.

For example, on input $$\langle 2,1 \rangle$$,

Remarks
 * The leftmost-innermost strategy is implemented in 225 computer languages on Rosetta Code.
 * For all $$m,n$$ the computation of $$A(m,n)$$ takes no more than $$(A(m,n) + 1)^m$$ steps.
 * pointed out that in the computation of $$\operatorname{A}(m,n)$$ the maximum length of the stack is $$\operatorname{A}(m,n)$$, as long as $$m>0$$.
 * Their own algorithm, inherently iterative, computes $$\operatorname{A}(m,n)$$ within $$\mathcal{O}(m \operatorname{A}(m,n))$$ time and within $$\mathcal{O}(m)$$ space.

TRS, based on iterated 1-ary function
The definition of the iterated 1-ary Ackermann functions leads to different reduction rules

\begin{array}{lll} \text{(r4)} & A(S(0),0,n)   & \rightarrow & S(n) \\ \text{(r5)} & A(S(0),S(m),n) & \rightarrow & A(S(n),m,S(0)) \\ \text{(r6)} & A(S(S(x)),m,n) & \rightarrow & A(S(0),m,A(S(x),m,n)) \end{array} $$

As function composition is associative, instead of rule r6 one can define

\begin{array}{lll} \text{(r7)} & A(S(S(x)),m,n) & \rightarrow & A(S(x),m,A(S(0),m,n)) \end{array} $$

Like in the previous section the computation of $$\operatorname{A}^1_m(n)$$ can be implemented with a stack.

Initially the stack contains the three elements $$\langle 1,m,n \rangle$$.

Then repeatedly the three top elements are replaced according to the rules

\begin{array}{lllllllll} \text{(r4)} & 1  &, 0   &, n & \rightarrow & (n+1) \\ \text{(r5)} & 1  &, (m+1) &, n & \rightarrow & (n+1) &, m &, 1 \\ \text{(r6)} & (x+2) &, m  &, n & \rightarrow & 1 &, m &, (x+1) &, m &, n \\ \end{array} $$

Schematically, starting from $$\langle 1, m,n \rangle$$: WHILE stackLength <> 1 {   POP 3 elements; PUSH 1 or 3 or 5 elements, applying the rules r4, r5, r6; }

Example

On input $$\langle 1,2,1 \rangle$$ the successive stack configurations are
 * $$\begin{align}

& \underline{1,2,1} \rightarrow_{r5} \underline{2,1,1} \rightarrow_{r6} 1,1,\underline{1,1,1} \rightarrow_{r5} 1,1,\underline{2,0,1} \rightarrow_{r6} 1,1,1,0,\underline{1,0,1} \\ & \rightarrow_{r4} 1,1,\underline{1,0,2} \rightarrow_{r4} \underline{1,1,3} \rightarrow_{r5} \underline{4,0,1} \rightarrow_{r6} 1,0,\underline{3,0,1} \rightarrow_{r6} 1,0,1,0,\underline{2,0,1} \\ & \rightarrow_{r6} 1,0,1,0,1,0,\underline{1,0,1} \rightarrow_{r4} 1,0,1,0,\underline{1,0,2} \rightarrow_{r4} 1,0,\underline{1,0,3} \rightarrow_{r4} \underline{1,0,4} \rightarrow_{r4} 5 \end{align}$$

The corresponding equalities are
 * $$\begin{align}

& A_2(1) = A^2_1(1) = A_1(A_1(1)) = A_1(A^2_0(1)) = A_1(A_0(A_0(1))) \\ & = A_1(A_0(2)) = A_1(3) = A^4_0(1) = A_0(A^3_0(1)) = A_0(A_0(A^2_0(1))) \\ & = A_0(A_0(A_0(A_0(1)))) = A_0(A_0(A_0(2))) = A_0(A_0(3)) = A_0(4) = 5 \end{align}$$

When reduction rule r7 is used instead of rule r6, the replacements in the stack will follow
 * $$\begin{array}{lllllllll}

\text{(r7)} & (x+2) &, m  &, n & \rightarrow & (x+1) &, m &, 1 &, m &, n \end{array}$$

The successive stack configurations will then be
 * $$\begin{align}

& \underline{1,2,1} \rightarrow_{r5} \underline{2,1,1} \rightarrow_{r7} 1,1,\underline{1,1,1} \rightarrow_{r5} 1,1,\underline{2,0,1} \rightarrow_{r7} 1,1,1,0,\underline{1,0,1} \\ & \rightarrow_{r4} 1,1,\underline{1,0,2} \rightarrow_{r4} \underline{1,1,3} \rightarrow_{r5} \underline{4,0,1} \rightarrow_{r7} 3,0,\underline{1,0,1} \rightarrow_{r4} \underline{3,0,2} \\ & \rightarrow_{r7} 2,0,\underline{1,0,2} \rightarrow_{r4} \underline{2,0,3} \rightarrow_{r7} 1,0,\underline{1,0,3} \rightarrow_{r4} \underline{1,0,4} \rightarrow_{r4} 5 \end{align}$$

The corresponding equalities are
 * $$\begin{align}

& A_2(1) = A^2_1(1) = A_1(A_1(1)) = A_1(A^2_0(1)) = A_1(A_0(A_0(1))) \\ & = A_1(A_0(2)) = A_1(3) = A^4_0(1) = A^3_0(A_0(1)) = A^3_0(2) \\ & = A^2_0(A_0(2)) = A^2_0(3) = A_0(A_0(3)) = A_0(4) = 5 \end{align}$$

Remarks
 * On any given input the TRSs presented so far converge in the same number of steps. They also use the same reduction rules (in this comparison the rules r1, r2, r3 are considered "the same as" the rules r4, r5, r6/r7 respectively). For example, the reduction of $$A(2,1)$$ converges in 14 steps: 6 × r1, 3 × r2, 5 × r3. The reduction of $$A_2(1)$$ converges in the same 14 steps: 6 × r4, 3 × r5, 5 × r6/r7. The TRSs differ in the order in which the reduction rules are applied.
 * When $$A_{i}(n)$$ is computed following the rules {r4, r5, r6}, the maximum length of the stack stays below $$2 \times A(i,n)$$. When reduction rule r7 is used instead of rule r6, the maximum length of the stack is only $$2(i+2)$$. The length of the stack reflects the recursion depth. As the reduction according to the rules {r4, r5, r7} involves a smaller maximum depth of recursion, this computation is more efficient in that respect.

TRS, based on hyperoperators
As — or  — showed explicitly, the Ackermann function can be expressed in terms of the hyperoperation sequence:
 * $$A(m,n) = \begin{cases}

n+1 & m=0 \\ 2[m](n+3) - 3 & m>0 \\ \end{cases}$$

or, after removal of the constant 2 from the parameter list, in terms of Buck's function
 * $$ = \begin{cases}

n+1 & m=0 \\ F(m,n+3) - 3 & m>0 \\ \end{cases}$$ Buck's function $$\operatorname{F}(m,n) = 2[m]n$$, a variant of Ackermann function by itself, can be computed with the following reduction rules:

\begin{array}{lll} \text{(b1)} & F(S(0),0,n)         & \rightarrow & S(n) \\ \text{(b2)} & F(S(0),S(0),0)      & \rightarrow & S(S(0)) \\ \text{(b3)} & F(S(0),S(S(0)),0)   & \rightarrow & 0 \\ \text{(b4)} & F(S(0),S(S(S(m))),0) & \rightarrow & S(0) \\ \text{(b5)} & F(S(0),S(m),S(n))   & \rightarrow & F(S(n),m,F(S(0),S(m),0)) \\ \text{(b6)} & F(S(S(x)),m,n)      & \rightarrow & F(S(0),m,F(S(x),m,n)) \end{array} $$ Instead of rule b6 one can define the rule

\begin{array}{lll} \text{(b7)} & F(S(S(x)),m,n)      & \rightarrow & F(S(x),m,F(S(0),m,n)) \end{array} $$ To compute the Ackermann function it suffices to add three reduction rules

\begin{array}{lll} \text{(r8)} & A(0,n)	     & \rightarrow & S(n) \\ \text{(r9)} & A(S(m),n)     & \rightarrow & P(F(S(0),S(m),S(S(S(n))))) \\ \text{(r10)} & P(S(S(S(m)))) & \rightarrow & m \\ \end{array} $$

These rules take care of the base case A(0,n), the alignment (n+3) and the fudge (-3).

Example

Compute $$A(2,1) \rightarrow_{*} 5$$

The matching equalities are
 * when the TRS with the reduction rule $$\text{b6}$$ is applied:
 * $$\begin{align}

& A(2,1) +3 = F(2,4) = \dots = F^6(0,2) = F(0,F^5(0,2)) = F(0,F(0,F^4(0,2))) \\ & = F(0,F(0,F(0,F^3(0,2)))) = F(0,F(0,F(0,F(0,F^2(0,2))))) = F(0,F(0,F(0,F(0,F(0,F(0,2)))))) \\ & = F(0,F(0,F(0,F(0,F(0,3))))) = F(0,F(0,F(0,F(0,4)))) = F(0,F(0,F(0,5))) = F(0,F(0,6)) = F(0,7) = 8 \end{align}$$
 * when the TRS with the reduction rule $$\text{b7}$$ is applied:
 * $$\begin{align}

& A(2,1) +3 = F(2,4) = \dots = F^6(0,2) = F^5(0,F(0,2)) = F^5(0,3) = F^4(0,F(0,3)) = F^4(0,4) \\ & = F^3(0,F(0,4)) = F^3(0,5) = F^2(0,F(0,5)) = F^2(0,6) = F(0,F(0,6)) = F(0,7) = 8 \end{align}$$ Remarks
 * The computation of $$\operatorname{A}_{i}(n)$$ according to the rules {b1 - b5, b6, r8 - r10} is deeply recursive. The maximum depth of nested $$F$$s is $$A(i,n)+1$$. The culprit is the order in which iteration is executed: $$F^{n+1}(x) = F(F^{n}(x))$$. The first $$F$$ disappears only after the whole sequence is unfolded.
 * The computation according to the rules {b1 - b5, b7, r8 - r10} is more efficient in that respect. The iteration $$F^{n+1}(x) = F^{n}(F(x))$$ simulates the repeated loop over a block of code. The nesting is limited to $$(i+1)$$, one recursion level per iterated function. showed this correspondence.
 * These considerations concern the recursion depth only. Either way of iterating leads to the same number of reduction steps, involving the same rules (when the rules b6 and b7 are considered "the same"). The reduction of $$A(2,1)$$ for instance converges in 35 steps: 12 × b1, 4 × b2, 1 × b3, 4 × b5, 12 × b6/b7, 1 × r9, 1 × r10. The modus iterandi only affects the order in which the reduction rules are applied.
 * A real gain of execution time can only be achieved by not recalculating subresults over and over again. Memoization is an optimization technique where the results of function calls are cached and returned when the same inputs occur again. See for instance . published a cunning algorithm which computes $$A(i,n)$$ within $$\mathcal{O}(i A(i,n))$$ time and within $$\mathcal{O}(i)$$ space.

Huge numbers
To demonstrate how the computation of $$A(4, 3)$$ results in many steps and in a large number:

 * $$\begin{align}

A(4, 3) & \rightarrow A(3, A(4, 2)) \\ & \rightarrow A(3, A(3, A(4, 1))) \\ & \rightarrow A(3, A(3, A(3, A(4, 0)))) \\ & \rightarrow A(3, A(3, A(3, A(3, 1)))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(3, 0))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(2, 1))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, A(2, 0)))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, A(1, 1)))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0))))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1))))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, A(0, 2)))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(1, 3))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(1, 2)))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1))))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0)))))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1)))))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2)) )) )) ) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, A(0, 3)))))) \\ & \rightarrow A(3, A(3, A(3, A(2, A(0, 4))))) \\ & \rightarrow A(3, A(3, A(3, A(2, 5)))) \\ & \qquad\vdots \\ & \rightarrow A(3, A(3, A(3, 13))) \\ & \qquad\vdots \\ & \rightarrow A(3, A(3, 65533)) \\ &\qquad\vdots \\ & \rightarrow A(3, 2^{65536} - 3) \\ &\qquad\vdots \\ & \rightarrow 2^{2^{65536}} - 3. \\ \end{align}$$

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:

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:

General remarks

 * It may not be immediately obvious that the evaluation of $$A(m, n)$$ 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.
 * For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n (at most exponentially). For $$m\geq 4$$, however, it grows much more quickly; even $$A(4,2)$$ is about 2.00353, and the decimal expansion of $$A(4, 3)$$ is very large by any typical measure, about 2.12004.
 * An interesting aspect 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 above.
 * A single-argument version $$f(n)=A(n,n)$$ 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 $$f_{\omega}(n)$$ 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.

Not primitive recursive
The Ackermann function grows faster than any primitive recursive function and therefore is not itself primitive recursive. The sketch of the proof is this: a primitive recursive function defined using up to k recursions must grow slower than $$f_{k+1}(n)$$, the (k+1)-th function in the fast-growing hierarchy, but the Ackermann function grows at least as fast as $$f_\omega(n)$$.

Specifically, one shows that for every primitive recursive function $$f(x_1,\ldots,x_n)$$ there exists a non-negative integer $$t$$ such that for all non-negative integers $$x_1,\ldots,x_n$$,

 * $$f(x_1,\ldots,x_n)<A(t,\max_i x_i).$$

Once this is established, it follows that $$A$$ itself is not primitive recursive, since otherwise putting $$x_1=x_2=t$$ would lead to the contradiction $$A(t,t)<A(t,t).$$

The proof proceeds as follows: define the class $$\mathcal{A}$$ of all functions that grow slower than the Ackermann function

 * $$\mathcal{A}=\left\{ f\,\bigg|\,\exists t\ \forall x_1\cdots \forall x_n:\ f(x_1,\ldots,x_n)<A(t, \max_i x_i) \right\} $$

and show that $$\mathcal{A}$$ contains all primitive recursive functions. The latter is achieved by showing that $$\mathcal{A}$$ contains the constant functions, the successor function, the projection functions and that it is closed under the operations of function composition and primitive recursion.

Use in computational complexity
The Ackermann function appears in the time complexity of some algorithms, such as vector addition systems and Petri net reachability, thus showing they are computationally infeasible for large instances. The inverse of the Ackerman function appears in some time complexity results.

Inverse
Since the function f(n) = A(n, n) considered above grows very rapidly, its inverse function, f, grows very slowly. This inverse Ackermann function f−1 is usually denoted by α. In fact, α(n) is less than 5 for any practical input size n, since A(4, 4) is on the order of $$2^{2^{2^{2^{16}}}}$$.

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 $$\lfloor x \rfloor$$ is the floor function:

 * $$\alpha(m,n) = \min\{i \geq 1 : A(i,\lfloor m/n \rfloor) \geq \log_2 n\}.$$

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 α(m, n) exist; for example, log2 n 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 Dragoș 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.

title: Ackermann function


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The Antarctic comprises the continent of Antarctica, the Kerguelen Plateau, and other island territories located on the Antarctic Plate or south of the Antarctic Convergence. The Antarctic region includes the ice shelves, waters, and all the island territories in the Southern Ocean situated south of the Antarctic Convergence, a zone approximately 32 to 48 km wide and varying in latitude seasonally. The region covers some 20 percent of the Southern Hemisphere, of which 5.5 percent (14 million km2) is the surface area of the Antarctica continent itself. All of the land and ice shelves south of 60°S latitude are administered under the Antarctic Treaty System.

Biogeographically, the Antarctic realm is one of eight biogeographic realms on Earth's land surface. Climate change in Antarctica is particularly important because the melting of the Antarctic ice sheet has a high potential to add to the global sea level rise. Further, this melting also disrupts the flow of Southern Ocean overturning circulation, which would have significant effects on the local climate and marine ecosystem functioning.

Geography

As defined by the Antarctic Treaty System, the Antarctic region is everything south of the 60°S latitude. The Treaty area covers Antarctica and the archipelagos of the Balleny Islands, Peter I Island, Scott Island, the South Orkney Islands, and the South Shetland Islands. However, this area does not include the Antarctic Convergence, a transition zone where the cold waters of the Southern Ocean collide with the warmer waters of the north, forming a natural border to the region. Because the Convergence changes seasonally, the Convention for the Conservation of Antarctic Marine Living Resources approximates the Convergence line by joining specified points along parallels of latitude and meridians of longitude. The implementation of the convention is managed through an international commission headquartered in Hobart, Australia, by an efficient system of annual fishing quotas, licenses, and international inspectors on the fishing vessels, as well as satellite surveillance.

The islands situated between 60°S latitude parallel to the south and the Antarctic Convergence to the north and their respective 200 nmi exclusive economic zones fall under the national jurisdiction of the countries that possess them: South Georgia and the South Sandwich Islands (United Kingdom), Bouvet Island (Norway), and Heard and McDonald Islands (Australia).

Kerguelen Islands (France; also an EU Overseas territory) are situated in the Antarctic Convergence area, while the Isla Grande de Tierra del Fuego, Falkland Islands, Isla de los Estados, Hornos Island with Cape Horn, Diego Ramírez Islands, Campbell Island, Macquarie Island, Amsterdam and Saint Paul Islands, Crozet Islands, Prince Edward Islands, Gough Island, and Tristan da Cunha group remain north of the Convergence and thus outside the Antarctic region.

Antarctica
A variety of animals live in Antarctica for at least some of the year, including: Most of the Antarctica continent is permanently covered by ice and snow, leaving less than 1 percent of the land exposed. There are only two species of flowering plant, Antarctic hair grass and Antarctic pearlwort, but a range of mosses, liverworts, lichens and macrofungi.
 * Seals
 * Penguins
 * South Georgia pipits
 * Albatrosses
 * Antarctic petrels
 * Whales
 * Fish, such as Antarctic icefish, Antarctic toothfish
 * Squid, including the colossal squid
 * Antarctic krill

Sub-Antarctic Islands
Biodiversity among terrestrial flora and fauna is low on the islands: studies have theorized that the harsh climate was a major contributor towards species richness, but multiple correlations have been found with area, temperature, remoteness of islands, and food chain stability. For example, herbivorous insects are poor in number due to low plant richness, and likewise, indigenous bird numbers are related to insects, which are a major food source.
 * 🇦🇷 Isla de los Estados (Argentina)
 * 🇨🇱 Isla Grande de Tierra del Fuego (Chile)

Conservation
The Antarctic hosts the world's largest protected area comprising 1.07 million km2, the South Georgia and the South Sandwich Islands Marine Protection Area created in 2012. The latter exceeds the surface area of another vast protected territory, the Greenland National Park's 972,000 km2. (While the Ross Sea Marine Protection Area established in 2016 is still larger at 1.55 million km2, its protection is set to expire in 35 years. ) To protect the area, all Antarctic ships over 500 tonnes are subject to mandatory regulations under the Polar Code, adopted by the International Maritime Organization (in force since 1 January 2017).

People
The first recorded sighting of Antarctica is credited to the Spaniard Gabriel de Castilla, who reported seeing distant southern snow-capped mountains in 1603. The first Antarctic land discovered was the island of South Georgia, visited by the English merchant Anthony de la Roché in 1675.

Although such myths and speculation about a Terra Australis ("Southern Land") date back to antiquity, the first confirmed sighting of the continent of Antarctica is commonly accepted to have occurred in 1820 by the Russian expedition of Fabian Gottlieb von Bellingshausen and Mikhail Lazarev on Vostok and Mirny.

The Australian James Kerguelen Robinson (1859–1914) was the first human born in the Antarctic, on board the sealing ship Offley in the Gulf of Morbihan (Royal Sound then), Kerguelen Island on 11 March 1859. The first human born and raised on an Antarctic island was Solveig Gunbjørg Jacobsen born on 8 October 1913 in Grytviken, South Georgia. Emilio Marcos Palma (born 7 January 1978) is an Argentine man who was the first documented person born on the continent of Antarctica at the Esperanza Base. His father, Captain Jorge Palma, was head of the Argentine Army detachment at the base. While ten people have been born in Antarctica since, Palma's birthplace remains the southernmost. In late 1977, Silvia Morella de Palma, who was then seven months pregnant, was airlifted to Esperanza Base, in order to complete her pregnancy in the base. The airlift was a part of the Argentine solutions to the sovereignty dispute over territory in Antarctica. Emilio was automatically granted Argentine citizenship by the government since his parents were both Argentine citizens, and he was born in the claimed Argentine Antarctica. Palma can be considered to be the first native Antarctican. The Antarctic region had no indigenous population when first discovered, and its present inhabitants comprise a few thousand transient scientific and other personnel working on tours of duty at the several dozen research stations maintained by various countries. However, the region is visited by more than 40,000 tourists annually, the most popular destinations being the Antarctic Peninsula area (especially the South Shetland Islands) and South Georgia Island.

In December 2009, the growth of tourism, with consequences for both the ecology and the safety of the travellers in its great and remote wilderness, was noted at a conference in New Zealand by experts from signatories to the Antarctic Treaty. The definitive results of the conference were presented at the Antarctic Treaty states' meeting in Uruguay in May 2010.

Time zones
Because Antarctica surrounds the South Pole, it is theoretically located in all time zones. For practical purposes, time zones are usually based on territorial claims or the time zone of a station's owner country or supply base.

North of 60°S latitude
• 🇧🇻 Bouvet Island (Norway)

• 🇭🇲 Heard Island and McDonald Islands (Australia)

• * Heard Island

• * McDonald Islands

• 🇹🇫 Kerguelen Islands (France)

• South Georgia and the South Sandwich Islands (United Kingdom)

• * South Georgia Islands

• ** Shag Rocks

• ** South Georgia

• * South Sandwich Islands

South of 60°S latitude
• Balleny Islands (Antarctic Treaty System)

•  Peter I Island (Antarctic Treaty System)

•  Scott Island (Antarctic Treaty System)

•  South Orkney Islands (Antarctic Treaty System)

•  South Shetland Islands (Antarctic Treaty System)

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title: Antarctic