Ackermann function: Difference between revisions

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In [[computability theory]], the '''Ackermann function''' or '''Ackermann-Péter function''' is a simple example of a [[computable function]] that is not [[Primitive recursive function|primitive recursive]]. The set of primitive recursive functions is a [[subset]] of the set of general recursive functions. Ackermann's function is an example that shows that the former is a [[strict subset]] of the latter.
In [[computability theory]], the '''Ackermann function''' or '''Ackermann-Péter function''' is a simple example of a [[computable function]] that is not [[Primitive recursive function|primitive recursive]]. The set of primitive recursive functions is a [[subset]] of the set of general recursive functions. Ackermann's function is an example that shows that the former is a [[strict subset]] of the latter.
<ref>{{cite journal | author=Wilhelm Ackermann | journal=[[Mathematische Annalen]] | title=''Zum Hilbertschen Aufbau der reellen Zahlen'' | year=1928 | volume=99 | pages=118&ndash;133 | doi=10.1007/BF01459088}}</ref>.


==Definiton==
The Ackermann function is defined [[recursion|recursively]] for non-negative integers ''m'' and ''n'' as follows::
The Ackermann function is defined [[recursion|recursively]] for non-negative integers ''m'' and ''n'' as follows::


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</math>
</math>


==Rapid growth==
Its value grows rapidly; even for small inputs, for example A(''4,2'') contains 19,729 decimal digits <ref>[http://www.kosara.net/thoughts/ackermann42.html Decimal expansion of A(4,2)]</ref>.
Its value grows rapidly; even for small inputs, for example A(''4,2'') contains 19,729 decimal digits <ref>[http://www.kosara.net/thoughts/ackermann42.html Decimal expansion of A(4,2)]</ref>.


==Holomorphic extensions==
The lowest Ackermann functions allow the extension to the complex values of the second argument. In particular,
:<math> A(0,z) =z+1</math>
:<math> A(1,z) =z+2=2+(n\!+\!3)-3</math>
:<math> A(1,z) =2z+3=2\!\cdot\!(n\!+\!3)-3</math>
The 3th Ackermann function is related to the exponential on base 2 through
:<math> A(3,z) = \exp_2(z\!+\!3)-3=2^{z+3}-3</math>
The 4th Ackermann function is related to [[tetration]] on base 2 through
:<math> A(4,z)=\mathrm{tet}_2(z+3)-3</math>
which allows its holomorphic extension for the complex values of the second argument.
<ref name="k">
D. Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008,
http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
</ref>
For <math>n>4</math> no holomorphic extension of <math>A(n,z)</math> to complex values of <math>z</math> is yet reported.
==References==
==References==
<references/>
<references/>

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In computability theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. The set of primitive recursive functions is a subset of the set of general recursive functions. Ackermann's function is an example that shows that the former is a strict subset of the latter. [1].

Definiton

The Ackermann function is defined recursively for non-negative integers m and n as follows::

Rapid growth

Its value grows rapidly; even for small inputs, for example A(4,2) contains 19,729 decimal digits [2].

Holomorphic extensions

The lowest Ackermann functions allow the extension to the complex values of the second argument. In particular,

The 3th Ackermann function is related to the exponential on base 2 through

The 4th Ackermann function is related to tetration on base 2 through

which allows its holomorphic extension for the complex values of the second argument. [3]

For no holomorphic extension of to complex values of is yet reported.

References

  1. Wilhelm Ackermann (1928). "Zum Hilbertschen Aufbau der reellen Zahlen". Mathematische Annalen 99: 118–133. DOI:10.1007/BF01459088. Research Blogging.
  2. Decimal expansion of A(4,2)
  3. D. Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008, http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf