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=118–133 | 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/> | ||
Revision as of 06:21, 11 July 2009
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
- ↑ Wilhelm Ackermann (1928). "Zum Hilbertschen Aufbau der reellen Zahlen". Mathematische Annalen 99: 118–133. DOI:10.1007/BF01459088. Research Blogging.
- ↑ Decimal expansion of A(4,2)
- ↑ D. Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008, http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf