Editing Ackermann function (section)

==Definition==
=== Definition: as m-ary function ===
Ackermann's original three-argument function <math>\varphi(m, n, p)</math> is defined [[recursion|recursively]] as follows for nonnegative integers <math>m,n,</math> and <math>p</math>:

<math display="block">\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}</math>

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 <math>m</math> and <math>n</math> as follows:

<math display="block"> 
\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}
</math>

The Ackermann function has also been expressed in relation to the [[Hyperoperation|hyperoperation sequence]]:{{sfn|Sundblad|1971}}{{sfn|Porto|Matos|1980}}

<math display="block">A(m,n) = \begin{cases}
 n+1 & m=0 \\
 2[m](n+3)-3 & m>0 \\
\end{cases}</math>

or, written in [[Knuth's up-arrow notation]] (extended to integer indices <math>\geq -2</math>):

<math display="block">A(m,n) = \begin{cases}
 n+1 & m=0 \\
 2\uparrow^{m-2} (n+3) - 3 & m>0 \\
 \end{cases}</math>

or, equivalently, in terms of Buck's function F:{{sfn|Buck|1963}}  

<math display="block">A(m,n) = \begin{cases}
 n+1 & m=0 \\
 F(m,n+3) - 3 & m>0 \\
\end{cases}</math>

=== Definition: as iterated 1-ary function ===
Define <math>f^{n}</math> as the ''n''-th iterate of <math>f</math>:

<math display="block">\begin{array}{rll}
f^{0}(x) & = & x \\
f^{n+1}(x) & = & f(f^{n}(x))
\end{array}</math>

[[Iterated function|Iteration]] is the process of composing a function with itself a certain number of times. [[Function composition]] is an [[associative]] operation, so <math>f(f^{n}(x)) = f^{n}(f(x))</math>.

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

The function then becomes a sequence <math>\operatorname{A}_0, \operatorname{A}_1, \operatorname{A}_2, ...</math> of unary<ref group="n" name="letop4">'[[Currying|curried]]'</ref> functions, defined from [[Iterated function|iteration]]:

<math display="block"> 
\begin{array}{lcl}
\operatorname{A}_{0}(n) & = & n+1 \\
\operatorname{A}_{m+1}(n) & = & \operatorname{A}_{m}^{n+1}(1) \\
\end{array}
</math>