Ackermann function

This article is about the mathematical function. For other uses, see Ackermann (disambiguation).

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest ^[1] 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 ^[2] 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. This function is defined from the recurrence relation ${\displaystyle \operatorname {A} (m+1,n+1)=\operatorname {A} (m,\operatorname {A} (m+1,n))}$ with appropriate base cases. Its value grows very rapidly; for example, ${\displaystyle \operatorname {A} (4,2)}$ results in ${\displaystyle 2^{65536}-3}$, an integer with 19,729 decimal digits. ^[3]

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 ^[4] 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 ${\displaystyle \varphi }$ (from Greek, the letter phi ). Ackermann's three-argument function, ${\displaystyle \varphi (m,n,p)}$, is defined such that for ${\displaystyle p=0,1,2}$, it reproduces the basic operations of addition, multiplication, and exponentiation as

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

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

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

(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, ^[5] 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. ^{[2] [6]}

Rózsa Péter ^[7] and Raphael Robinson ^[8] later developed a two-variable version of the Ackermann function that became preferred by almost all authors.

The generalized hyperoperation sequence, e.g. ${\displaystyle G(m,a,b)=a[m]b}$, is a version of the Ackermann function as well. ^[9]

In 1963 R.C. Buck based an intuitive two-variable ^{[n 1]} variant ${\displaystyle \operatorname {F} }$ on the hyperoperation sequence: ^{[10] [11]}

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

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

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

Many other versions of Ackermann function have been investigated. ^{[12] [13]}

Definition

Definition: as m-ary function

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

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

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 ${\displaystyle m}$ and ${\displaystyle n}$ as follows:

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

The Ackermann function has also been expressed in relation to the hyperoperation sequence: ^{[14] [15]}

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

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

$${\displaystyle A(m,n)={\begin{cases}n+1&m=0\\2\uparrow ^{m-2}(n+3)-3&m>0\\\end{cases}}}$$

or, equivalently, in terms of Buck's function F: ^[10]

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

Definition: as iterated 1-ary function

Define ${\displaystyle f^{n}}$ as the n-th iterate of ${\displaystyle f}$:

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

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

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

The function then becomes a sequence ${\displaystyle \operatorname {A} _{0},\operatorname {A} _{1},\operatorname {A} _{2},...}$ of unary ^{[n 2]} functions, defined from iteration:

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

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 ^{[16] [17]}

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

Example

Compute ${\displaystyle A(1,2)\rightarrow _{*}4}$

The reduction sequence is ^{[n 3]}

Leftmost-outermost (one-step) strategy:	Leftmost-innermost (one-step) strategy:
${\displaystyle {\underline {A(S(0),S(S(0)))}}}$	${\displaystyle {\underline {A(S(0),S(S(0)))}}}$
${\displaystyle \rightarrow _{r3}{\underline {A(0,A(S(0),S(0))}})}$	${\displaystyle \rightarrow _{r3}A(0,{\underline {A(S(0),S(0))}})}$
${\displaystyle \rightarrow _{r1}S({\underline {A(S(0),S(0))}})}$	${\displaystyle \rightarrow _{r3}A(0,A(0,{\underline {A(S(0),0)}}))}$
${\displaystyle \rightarrow _{r3}S({\underline {A(0,A(S0,0))}})}$	${\displaystyle \rightarrow _{r2}A(0,A(0,{\underline {A(0,S(0))}}))}$
${\displaystyle \rightarrow _{r1}S(S({\underline {A(S(0),0)}}))}$	${\displaystyle \rightarrow _{r1}A(0,{\underline {A(0,S(S(0)))}})}$
${\displaystyle \rightarrow _{r2}S(S({\underline {A(0,S(0))}}))}$	${\displaystyle \rightarrow _{r1}{\underline {A(0,S(S(S(0))))}}}$
${\displaystyle \rightarrow _{r1}S(S(S(S(0))))}$	${\displaystyle \rightarrow _{r1}S(S(S(S(0))))}$

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

Then repeatedly the two top elements are replaced according to the rules ^{[n 4]}

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

Schematically, starting from ${\displaystyle \langle m,n\rangle }$:

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

The pseudocode is published in Grossman & Zeitman (1988).

For example, on input ${\displaystyle \langle 2,1\rangle }$,

the stack configurations	reflect the reduction [n 5]
${\displaystyle {\underline {2,1}}}$	${\displaystyle {\underline {A(2,1)}}}$
${\displaystyle \rightarrow 1,{\underline {2,0}}}$	${\displaystyle \rightarrow _{r1}A(1,{\underline {A(2,0)}})}$
${\displaystyle \rightarrow 1,{\underline {1,1}}}$	${\displaystyle \rightarrow _{r2}A(1,{\underline {A(1,1)}})}$
${\displaystyle \rightarrow 1,0,{\underline {1,0}}}$	${\displaystyle \rightarrow _{r3}A(1,A(0,{\underline {A(1,0)}}))}$
${\displaystyle \rightarrow 1,0,{\underline {0,1}}}$	${\displaystyle \rightarrow _{r2}A(1,A(0,{\underline {A(0,1)}}))}$
${\displaystyle \rightarrow 1,{\underline {0,2}}}$	${\displaystyle \rightarrow _{r1}A(1,{\underline {A(0,2)}})}$
${\displaystyle \rightarrow {\underline {1,3}}}$	${\displaystyle \rightarrow _{r1}{\underline {A(1,3)}}}$
${\displaystyle \rightarrow 0,{\underline {1,2}}}$	${\displaystyle \rightarrow _{r3}A(0,{\underline {A(1,2)}})}$
${\displaystyle \rightarrow 0,0,{\underline {1,1}}}$	${\displaystyle \rightarrow _{r3}A(0,A(0,{\underline {A(1,1)}}))}$
${\displaystyle \rightarrow 0,0,0,{\underline {1,0}}}$	${\displaystyle \rightarrow _{r3}A(0,A(0,A(0,{\underline {A(1,0)}})))}$
${\displaystyle \rightarrow 0,0,0,{\underline {0,1}}}$	${\displaystyle \rightarrow _{r2}A(0,A(0,A(0,{\underline {A(0,1)}})))}$
${\displaystyle \rightarrow 0,0,{\underline {0,2}}}$	${\displaystyle \rightarrow _{r1}A(0,A(0,{\underline {A(0,2)}}))}$
${\displaystyle \rightarrow 0,{\underline {0,3}}}$	${\displaystyle \rightarrow _{r1}A(0,{\underline {A(0,3)}})}$
${\displaystyle \rightarrow {\underline {0,4}}}$	${\displaystyle \rightarrow _{r1}{\underline {A(0,4)}}}$
${\displaystyle \rightarrow 5}$	${\displaystyle \rightarrow _{r1}5}$

Remarks

• The leftmost-innermost strategy is implemented in 225 computer languages on Rosetta Code.
• For all ${\displaystyle m,n}$ the computation of ${\displaystyle A(m,n)}$ takes no more than ${\displaystyle (A(m,n)+1)^{m}}$ steps. ^[18]
• Grossman & Zeitman (1988) pointed out that in the computation of ${\displaystyle \operatorname {A} (m,n)}$ the maximum length of the stack is ${\displaystyle \operatorname {A} (m,n)}$, as long as ${\displaystyle m>0}$.

  Their own algorithm, inherently iterative, computes ${\displaystyle \operatorname {A} (m,n)}$ within ${\displaystyle {\mathcal {O}}(m\operatorname {A} (m,n))}$ time and within ${\displaystyle {\mathcal {O}}(m)}$ space.

TRS, based on iterated 1-ary function

The definition of the iterated 1-ary Ackermann functions leads to different reduction rules

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

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

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

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

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

Then repeatedly the three top elements are replaced according to the rules ^{[n 4]}

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

Schematically, starting from ${\displaystyle \langle 1,m,n\rangle }$:

WHILE stackLength <> 1 {    POP 3 elements;    PUSH 1 or 3 or 5 elements, applying the rules r4, r5, r6; }

Example

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

$${\displaystyle {\begin{aligned}&{\underline {1,2,1}}\rightarrow _{r5}{\underline {2,1,1}}\rightarrow _{r6}1,1,{\underline {1,1,1}}\rightarrow _{r5}1,1,{\underline {2,0,1}}\rightarrow _{r6}1,1,1,0,{\underline {1,0,1}}\\&\rightarrow _{r4}1,1,{\underline {1,0,2}}\rightarrow _{r4}{\underline {1,1,3}}\rightarrow _{r5}{\underline {4,0,1}}\rightarrow _{r6}1,0,{\underline {3,0,1}}\rightarrow _{r6}1,0,1,0,{\underline {2,0,1}}\\&\rightarrow _{r6}1,0,1,0,1,0,{\underline {1,0,1}}\rightarrow _{r4}1,0,1,0,{\underline {1,0,2}}\rightarrow _{r4}1,0,{\underline {1,0,3}}\rightarrow _{r4}{\underline {1,0,4}}\rightarrow _{r4}5\end{aligned}}}$$

The corresponding equalities are

$${\displaystyle {\begin{aligned}&A_{2}(1)=A_{1}^{2}(1)=A_{1}(A_{1}(1))=A_{1}(A_{0}^{2}(1))=A_{1}(A_{0}(A_{0}(1)))\\&=A_{1}(A_{0}(2))=A_{1}(3)=A_{0}^{4}(1)=A_{0}(A_{0}^{3}(1))=A_{0}(A_{0}(A_{0}^{2}(1)))\\&=A_{0}(A_{0}(A_{0}(A_{0}(1))))=A_{0}(A_{0}(A_{0}(2)))=A_{0}(A_{0}(3))=A_{0}(4)=5\end{aligned}}}$$

When reduction rule r7 is used instead of rule r6, the replacements in the stack will follow

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

The successive stack configurations will then be

$${\displaystyle {\begin{aligned}&{\underline {1,2,1}}\rightarrow _{r5}{\underline {2,1,1}}\rightarrow _{r7}1,1,{\underline {1,1,1}}\rightarrow _{r5}1,1,{\underline {2,0,1}}\rightarrow _{r7}1,1,1,0,{\underline {1,0,1}}\\&\rightarrow _{r4}1,1,{\underline {1,0,2}}\rightarrow _{r4}{\underline {1,1,3}}\rightarrow _{r5}{\underline {4,0,1}}\rightarrow _{r7}3,0,{\underline {1,0,1}}\rightarrow _{r4}{\underline {3,0,2}}\\&\rightarrow _{r7}2,0,{\underline {1,0,2}}\rightarrow _{r4}{\underline {2,0,3}}\rightarrow _{r7}1,0,{\underline {1,0,3}}\rightarrow _{r4}{\underline {1,0,4}}\rightarrow _{r4}5\end{aligned}}}$$

The corresponding equalities are

$${\displaystyle {\begin{aligned}&A_{2}(1)=A_{1}^{2}(1)=A_{1}(A_{1}(1))=A_{1}(A_{0}^{2}(1))=A_{1}(A_{0}(A_{0}(1)))\\&=A_{1}(A_{0}(2))=A_{1}(3)=A_{0}^{4}(1)=A_{0}^{3}(A_{0}(1))=A_{0}^{3}(2)\\&=A_{0}^{2}(A_{0}(2))=A_{0}^{2}(3)=A_{0}(A_{0}(3))=A_{0}(4)=5\end{aligned}}}$$

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 ${\displaystyle A(2,1)}$ converges in 14 steps: 6 × r1, 3 × r2, 5 × r3. The reduction of ${\displaystyle A_{2}(1)}$ 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 ${\displaystyle A_{i}(n)}$ is computed following the rules {r4, r5, r6}, the maximum length of the stack stays below ${\displaystyle 2\times A(i,n)}$. When reduction rule r7 is used instead of rule r6, the maximum length of the stack is only ${\displaystyle 2(i+2)}$. 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, ^{[n 6]} this computation is more efficient in that respect.

TRS, based on hyperoperators

As Sundblad (1971) — or Porto & Matos (1980) — showed explicitly, the Ackermann function can be expressed in terms of the hyperoperation sequence:

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

or, after removal of the constant 2 from the parameter list, in terms of Buck's function

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

Buck's function ${\displaystyle \operatorname {F} (m,n)=2[m]n}$, ^[10] a variant of Ackermann function by itself, can be computed with the following reduction rules:

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

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

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

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

Example

Compute ${\displaystyle A(2,1)\rightarrow _{*}5}$

using reduction rule ${\displaystyle {\text{b7}}}$: [n 5]	using reduction rule ${\displaystyle {\text{b6}}}$: [n 5]
${\displaystyle {\underline {A(2,1)}}}$	${\displaystyle {\underline {A(2,1)}}}$
${\displaystyle \rightarrow _{r9}P({\underline {F(1,2,4)}})}$	${\displaystyle \rightarrow _{r9}P({\underline {F(1,2,4)}})}$
${\displaystyle \rightarrow _{b5}P(F(4,1,{\underline {F(1,2,0)}}))}$	${\displaystyle \rightarrow _{b5}P(F(4,1,{\underline {F(1,2,0)}}))}$
${\displaystyle \rightarrow _{b3}P({\underline {F(4,1,0)}})}$	${\displaystyle \rightarrow _{b3}P({\underline {F(4,1,0)}})}$
${\displaystyle \rightarrow _{b7}P(F(3,1,{\underline {F(1,1,0)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,{\underline {F(3,1,0)}}))}$
${\displaystyle \rightarrow _{b2}P({\underline {F(3,1,2)}})}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,1,{\underline {F(2,1,0)}})))}$
${\displaystyle \rightarrow _{b7}P(F(2,1,{\underline {F(1,1,2)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,1,F(1,1,{\underline {F(1,1,0)}}))))}$
${\displaystyle \rightarrow _{b5}P(F(2,1,F(2,0,{\underline {F(1,1,0)}})))}$	${\displaystyle \rightarrow _{b2}P(F(1,1,F(1,1,{\underline {F(1,1,2)}})))}$
${\displaystyle \rightarrow _{b2}P(F(2,1,{\underline {F(2,0,2)}}))}$	${\displaystyle \rightarrow _{b5}P(F(1,1,F(1,1,F(2,0,{\underline {F(1,1,0)}}))))}$
${\displaystyle \rightarrow _{b7}P(F(2,1,F(1,0,{\underline {F(1,0,2)}})))}$	${\displaystyle \rightarrow _{b2}P(F(1,1,F(1,1,{\underline {F(2,0,2)}})))}$
${\displaystyle \rightarrow _{b1}P(F(2,1,{\underline {F(1,0,3)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,1,F(1,0,{\underline {F(1,0,2)}}))))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(2,1,4)}})}$	${\displaystyle \rightarrow _{b1}P(F(1,1,F(1,1,{\underline {F(1,0,3)}})))}$
${\displaystyle \rightarrow _{b7}P(F(1,1,{\underline {F(1,1,4)}}))}$	${\displaystyle \rightarrow _{b1}P(F(1,1,{\underline {F(1,1,4)}}))}$
${\displaystyle \rightarrow _{b5}P(F(1,1,F(4,0,{\underline {F(1,1,0)}})))}$	${\displaystyle \rightarrow _{b5}P(F(1,1,F(4,0,{\underline {F(1,1,0)}})))}$
${\displaystyle \rightarrow _{b2}P(F(1,1,{\underline {F(4,0,2)}}))}$	${\displaystyle \rightarrow _{b2}P(F(1,1,{\underline {F(4,0,2)}}))}$
${\displaystyle \rightarrow _{b7}P(F(1,1,F(3,0,{\underline {F(1,0,2)}})))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,0,{\underline {F(3,0,2)}})))}$
${\displaystyle \rightarrow _{b1}P(F(1,1,{\underline {F(3,0,3)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,0,F(1,0,{\underline {F(2,0,2)}}))))}$
${\displaystyle \rightarrow _{b7}P(F(1,1,F(2,0,{\underline {F(1,0,3)}})))}$	${\displaystyle \rightarrow _{b6}P(F(1,1,F(1,0,F(1,0,F(1,0,{\underline {F(1,0,2)}})))))}$
${\displaystyle \rightarrow _{b1}P(F(1,1,{\underline {F(2,0,4)}}))}$	${\displaystyle \rightarrow _{b1}P(F(1,1,F(1,0,F(1,0,{\underline {F(1,0,3)}}))))}$
${\displaystyle \rightarrow _{b7}P(F(1,1,F(1,0,{\underline {F(1,0,4)}})))}$	${\displaystyle \rightarrow _{b1}P(F(1,1,F(1,0,{\underline {F(1,0,4)}})))}$
${\displaystyle \rightarrow _{b1}P(F(1,1,{\underline {F(1,0,5)}}))}$	${\displaystyle \rightarrow _{b1}P(F(1,1,{\underline {F(1,0,5)}}))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(1,1,6)}})}$	${\displaystyle \rightarrow _{b1}P({\underline {F(1,1,6)}})}$
${\displaystyle \rightarrow _{b5}P(F(6,0,{\underline {F(1,1,0)}}))}$	${\displaystyle \rightarrow _{b5}P(F(6,0,{\underline {F(1,1,0)}}))}$
${\displaystyle \rightarrow _{b2}P({\underline {F(6,0,2)}})}$	${\displaystyle \rightarrow _{b2}P({\underline {F(6,0,2)}})}$
${\displaystyle \rightarrow _{b7}P(F(5,0,{\underline {F(1,0,2)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,0,{\underline {F(5,0,2)}}))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(5,0,3)}})}$	${\displaystyle \rightarrow _{b6}P(F(1,0,F(1,0,{\underline {F(4,0,2)}})))}$
${\displaystyle \rightarrow _{b7}P(F(4,0,{\underline {F(1,0,3)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,0,F(1,0,F(1,0,{\underline {F(3,0,2)}}))))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(4,0,4)}})}$	${\displaystyle \rightarrow _{b6}P(F(1,0,F(1,0,F(1,0,F(1,0,{\underline {F(2,0,2)}})))))}$
${\displaystyle \rightarrow _{b7}P(F(3,0,{\underline {F(1,0,4)}}))}$	${\displaystyle \rightarrow _{b6}P(F(1,0,F(1,0,F(1,0,F(1,0,F(1,0,{\underline {F(1,0,2)}}))))))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(3,0,5)}})}$	${\displaystyle \rightarrow _{b1}P(F(1,0,F(1,0,F(1,0,F(1,0,{\underline {F(1,0,3)}})))))}$
${\displaystyle \rightarrow _{b7}P(F(2,0,{\underline {F(1,0,5)}}))}$	${\displaystyle \rightarrow _{b1}P(F(1,0,F(1,0,F(1,0,{\underline {F(1,0,4)}}))))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(2,0,6)}})}$	${\displaystyle \rightarrow _{b1}P(F(1,0,F(1,0,{\underline {F(1,0,5)}})))}$
${\displaystyle \rightarrow _{b7}P(F(1,0,{\underline {F(1,0,6)}}))}$	${\displaystyle \rightarrow _{b1}P(F(1,0,{\underline {F(1,0,6)}}))}$
${\displaystyle \rightarrow _{b1}P({\underline {F(1,0,7)}})}$	${\displaystyle \rightarrow _{b1}P({\underline {F(1,0,7)}})}$
${\displaystyle \rightarrow _{b1}{\underline {P(8)}}}$	${\displaystyle \rightarrow _{b1}{\underline {P(8)}}}$
${\displaystyle \rightarrow _{r10}5}$	${\displaystyle \rightarrow _{r10}5}$

The matching equalities are

• when the TRS with the reduction rule ${\displaystyle {\text{b6}}}$ is applied:

$${\displaystyle {\begin{aligned}&A(2,1)+3=F(2,4)=\dots =F^{6}(0,2)=F(0,F^{5}(0,2))=F(0,F(0,F^{4}(0,2)))\\&=F(0,F(0,F(0,F^{3}(0,2))))=F(0,F(0,F(0,F(0,F^{2}(0,2)))))=F(0,F(0,F(0,F(0,F(0,F(0,2))))))\\&=F(0,F(0,F(0,F(0,F(0,3)))))=F(0,F(0,F(0,F(0,4))))=F(0,F(0,F(0,5)))=F(0,F(0,6))=F(0,7)=8\end{aligned}}}$$

• when the TRS with the reduction rule ${\displaystyle {\text{b7}}}$ is applied:

$${\displaystyle {\begin{aligned}&A(2,1)+3=F(2,4)=\dots =F^{6}(0,2)=F^{5}(0,F(0,2))=F^{5}(0,3)=F^{4}(0,F(0,3))=F^{4}(0,4)\\&=F^{3}(0,F(0,4))=F^{3}(0,5)=F^{2}(0,F(0,5))=F^{2}(0,6)=F(0,F(0,6))=F(0,7)=8\end{aligned}}}$$Remarks

• The computation of ${\displaystyle \operatorname {A} _{i}(n)}$ according to the rules {b1 - b5, b6, r8 - r10} is deeply recursive. The maximum depth of nested ${\displaystyle F}$s is ${\displaystyle A(i,n)+1}$. The culprit is the order in which iteration is executed: ${\displaystyle F^{n+1}(x)=F(F^{n}(x))}$. The first ${\displaystyle F}$ 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 ${\displaystyle F^{n+1}(x)=F^{n}(F(x))}$ simulates the repeated loop over a block of code. ^{[n 7]} The nesting is limited to ${\displaystyle (i+1)}$, one recursion level per iterated function. Meyer & Ritchie (1967) 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 ${\displaystyle A(2,1)}$ 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 Ward (1993). Grossman & Zeitman (1988) published a cunning algorithm which computes ${\displaystyle A(i,n)}$ within ${\displaystyle {\mathcal {O}}(iA(i,n))}$ time and within ${\displaystyle {\mathcal {O}}(i)}$ space.

Huge numbers

To demonstrate how the computation of ${\displaystyle A(4,3)}$ results in many steps and in a large number: ^{[n 5]}

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

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)

n m	0	1	2	3	4	n
0	1	2	3	4	5	${\displaystyle n+1}$
1	2	3	4	5	6	${\displaystyle n+2=2+(n+3)-3}$
2	3	5	7	9	11	${\displaystyle 2n+3=2\cdot (n+3)-3}$
3	5	13	29	61	125	${\displaystyle 2^{(n+3)}-3}$
4	13	65533	265536 − 3	${\displaystyle {2^{2^{65536}}}-3}$	${\displaystyle {2^{2^{2^{65536}}}}-3}$	${\displaystyle {\begin{matrix}\underbrace {{2^{2}}^{{\cdot }^{{\cdot }^{{\cdot }^{2}}}}} _{n+3}-3\end{matrix}}}$
	${\displaystyle ={2^{2^{2}}}-3}$ ${\displaystyle =2\uparrow \uparrow 3-3}$	${\displaystyle ={2^{2^{2^{2}}}}-3}$ ${\displaystyle =2\uparrow \uparrow 4-3}$	${\displaystyle ={2^{2^{2^{2^{2}}}}}-3}$ ${\displaystyle =2\uparrow \uparrow 5-3}$	${\displaystyle ={2^{2^{2^{2^{2^{2}}}}}}-3}$ ${\displaystyle =2\uparrow \uparrow 6-3}$	${\displaystyle ={2^{2^{2^{2^{2^{2^{2}}}}}}}-3}$ ${\displaystyle =2\uparrow \uparrow 7-3}$
						${\displaystyle =2\uparrow \uparrow (n+3)-3}$
5	65533 ${\displaystyle =2\uparrow \uparrow (2\uparrow \uparrow 2)-3}$ ${\displaystyle =2\uparrow \uparrow \uparrow 3-3}$	${\displaystyle 2\uparrow \uparrow \uparrow 4-3}$	${\displaystyle 2\uparrow \uparrow \uparrow 5-3}$	${\displaystyle 2\uparrow \uparrow \uparrow 6-3}$	${\displaystyle 2\uparrow \uparrow \uparrow 7-3}$	${\displaystyle 2\uparrow \uparrow \uparrow (n+3)-3}$
6	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3-3}$	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 4-3}$	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 5-3}$	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 6-3}$	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow 7-3}$	${\displaystyle 2\uparrow \uparrow \uparrow \uparrow (n+3)-3}$
m	${\displaystyle (2\uparrow ^{m-2}3)-3}$	${\displaystyle (2\uparrow ^{m-2}4)-3}$	${\displaystyle (2\uparrow ^{m-2}5)-3}$	${\displaystyle (2\uparrow ^{m-2}6)-3}$	${\displaystyle (2\uparrow ^{m-2}7)-3}$	${\displaystyle (2\uparrow ^{m-2}(n+3))-3}$

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)

n m	0	1	2	3	4	n
0	0+1	1+1	2+1	3+1	4+1	n + 1
1	A(0, 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))
2	A(1, 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))
3	A(2, 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))
4	A(3, 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))
5	A(4, 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))
6	A(5, 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))

Properties

General remarks

• It may not be immediately obvious that the evaluation of ${\displaystyle A(m,n)}$ always terminates. However, the recursion is bounded because in each recursive application either ${\displaystyle m}$ decreases, or ${\displaystyle m}$ remains the same and ${\displaystyle n}$ decreases. Each time that ${\displaystyle n}$ reaches zero, ${\displaystyle m}$ decreases, so ${\displaystyle m}$ eventually reaches zero as well. (Expressed more technically, in each case the pair ${\displaystyle (m,n)}$ 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 ${\displaystyle m}$ decreases there is no upper bound on how much ${\displaystyle n}$ 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 ${\displaystyle m\geq 4}$, however, it grows much more quickly; even ${\displaystyle A(4,2)}$ is about 2.00353×10¹⁹⁷²⁸, and the decimal expansion of ${\displaystyle A(4,3)}$ is very large by any typical measure, about 2.12004×10^{6.03123×1019727}.
• 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 ${\displaystyle f(n)=A(n,n)}$ that increases both ${\displaystyle m}$ and ${\displaystyle n}$ 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 ${\displaystyle f(n)}$ is roughly comparable to ${\displaystyle f_{\omega }(n)}$ in the fast-growing hierarchy. This extreme growth can be exploited to show that ${\displaystyle f}$ 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. Proof sketch: primitive recursive function defined using up to k recursions must grow slower than ${\displaystyle f_{k+1}(n)}$, the (k+1)-th function in the fast-growing hierarchy, but the Ackermann function grows at least as fast as ${\displaystyle f_{\omega }(n)}$.

Specifically, one shows that, for every primitive recursive function ${\displaystyle f(x_{1},\ldots ,x_{n})}$, there exists a non-negative integer ${\displaystyle t}$, such that for all non-negative integers ${\displaystyle x_{1},\ldots ,x_{n}}$,$${\displaystyle f(x_{1},\ldots ,x_{n})<A(t,\max _{i}x_{i}).}$$Once this is established, it follows that ${\displaystyle A}$ itself is not primitive recursive, since otherwise putting ${\displaystyle x_{1}=x_{2}=t}$ would lead to the contradiction ${\displaystyle A(t,t)<A(t,t).}$

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

$${\displaystyle {\mathcal {A}}=\left\{f\,{\bigg |}\,\exists t\ \forall x_{1}\cdots \forall x_{n}:\ f(x_{1},\ldots ,x_{n})<A(t,\max _{i}x_{i})\right\}}$$

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

Inverse

Since the function f(n) = A(n, n) considered above grows very rapidly, its inverse function, f⁻¹, grows very slowly. This inverse Ackermann functionf⁻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 ${\displaystyle 2^{2^{2^{2^{16}}}}}$.

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 ${\displaystyle \lfloor x\rfloor }$ is the floor function:

$${\displaystyle \alpha (m,n)=\min\{i\geq 1:A(i,\lfloor m/n\rfloor )\geq \log _{2}n\}.}$$

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, log₂n 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. ^[19]

The inverse of the Ackermann function is primitive recursive, since it is graph primitive recursive, and it is upper bounded by a primitive recursive function. ^[20]

Usage

In computational complexity

The Ackermann function appears in the time complexity of some algorithms, ^[21] such as vector addition systems ^[22] and Petri net reachability, thus showing they are computationally infeasible for large instances. ^[23]

The inverse of the Ackermann function appears in some time complexity results. For instance, the disjoint-set data structure takes amortized time per operation proportional to the inverse Ackermann function, ^[24] and cannot be made faster within the cell-probe model of computational complexity. ^[25]

In discrete geometry

Certain problems in discrete geometry related to Davenport–Schinzel sequences have complexity bounds in which the inverse Ackermann function ${\displaystyle \alpha (n)}$ appears. For instance, for ${\displaystyle n}$ line segments in the plane, the unbounded face of the arrangement of the segments has complexity ${\displaystyle O(n\alpha (n))}$, and some systems of ${\displaystyle n}$ line segments have an unbounded face of complexity ${\displaystyle \Omega (n\alpha (n))}$. ^[26]

As a 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 ^[27] and, almost simultaneously, in 1971, by Yngve Sundblad. ^[14]

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. ^{[28] [29] [30]}

See also

• Computability theory
• Double recursion
• Fast-growing hierarchy
• Goodstein function
• Primitive recursive function
• Recursion (computer science)

Notes

1. with parameter order reversed
2. 'curried'
3. In each step the underlined redex is rewritten.
4. here: leftmost-innermost strategy!
5. For better readability
   S(0) is notated as 1,
   S(S(0)) is notated as 2,
   S(S(S(0))) is notated as 3,
   etc...
6. The maximum depth of recursion refers to the number of levels of activation of a procedure which exist during the deepest call of the procedure. Cornelius & Kirby (1975)
7. LOOP n+1 TIMES DO F

References

1. Monin & Hinchey 2003, p. 61.
2. Ackermann 1928.
3. "Decimal expansion of A(4,2)". kosara.net. 27 August 2000. Archived from the original on 20 January 2010.
4. Calude, Marcus & Tevy 1979.
5. Hilbert 1926, p. 185.
6. van Heijenoort 1977.
7. Péter 1935.
8. Robinson 1948.
9. Ritchie 1965, p. 1028.
10. Buck 1963.
11. Meeussen & Zantema 1992, p. 6.
12. Munafo 1999a.
13. Ritchie 1965.
14. Sundblad 1971.
15. Porto & Matos 1980.
16. Grossman & Zeitman 1988.
17. Paulson 2021.
18. Cohen 1987, p. 56, Proposition 3.16 (see in proof).
19. Pettie 2002.
20. Matos 2014.
21. Brubaker 2023.
22. Czerwiński & Orlikowski 2022.
23. Leroux 2022.
24. Tarjan 1975.
25. Fredman & Saks 1989.
26. Wiernik & Sharir 1988.
27. Vaida 1970.
28. Wichmann 1976.
29. Wichmann 1977.
30. Wichmann 1982.

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External links

• "Ackermann function". Encyclopedia of Mathematics . EMS Press. 2001 [1994].
• Weisstein, Eric W. "Ackermann function". MathWorld .
•  This article incorporates public domain material from Paul E. Black. "Ackermann's function". Dictionary of Algorithms and Data Structures . NIST.
• An animated Ackermann function calculator
• Aaronson, Scott (1999). "Who Can Name the Bigger Number?".
• Ackermann functions. Includes a table of some values.
• Brubaker, Ben (4 December 2023). "An Easy-Sounding Problem Yields Numbers Too Big for Our Universe".
• Munafo, Robert. "Large Numbers". describes several variations on the definition of A.
• Nivasch, Gabriel (October 2021). "Inverse Ackermann without pain". Archived from the original on 21 August 2007. Retrieved 18 June 2023.
• Seidel, Raimund. "Understanding the inverse Ackermann function" (PDF).
• The Ackermann function written in different programming languages, (on Rosetta Code)
• Smith, Harry J. "Ackermann's Function". Archived from the original on 26 October 2009.) Some study and programming.
• Wiernik, Ady; Sharir, Micha (1988). "Planar realizations of nonlinear Davenport–Schinzel sequences by segments". Discrete & Computational Geometry. 3 (1): 15–47. doi: 10.1007/BF02187894 . MR   0918177.