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In the field of [[recursion theory]], '''index sets''' describe classes of [[partial recursive function]]s, specifically they give all indices of functions in that class according to a fixed enumeration of partial recursive functions (a [[Gödel number]]ing). | |||
==Definition== | |||
Fix an enumeration of all partial recursive functions, or equivalently of [[recursively enumerable]] sets whereby the ''e''th such set is <math>W_{e}</math> and the ''e''th such function (whose domain is <math>W_{e}</math>) is <math>\phi_{e}</math>. | |||
Let <math>\mathcal{A}</math> be a class of partial recursive functions. If <math>A = \{x : \phi_{x} \in \mathcal{A} \}</math> then <math>A</math> is the '''index set''' of <math>\mathcal{A}</math>. In general <math>A</math> is an index set if for every <math>x,y \in \mathbb{N}</math> with <math>\phi_x \simeq \phi_y</math> (i.e. they index the same function), we have <math>x \in A \leftrightarrow y \in A</math>. Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index. | |||
==Index sets and [[Rice's theorem]]== | |||
Most index sets are incomputable (non-recursive) aside from two trivial exceptions. This is stated in '''Rice's theorem''': | |||
<blockquote>Let <math>\mathcal{C}</math> be a class of partial recursive functions with index set <math>C</math>. Then <math>C</math> is recursive if and only if <math>C</math> is empty, or <math>C</math> is all of <math>\omega</math>.</blockquote> | |||
where <math>\omega</math> is the set of [[natural numbers]], including [[zero]]. | |||
Rice's theorem says "any nontrivial property of partial recursive functions is undecidable"<ref name=odifreddi>{{cite book | title=Classical Recursion Theory, Volume 1| author=Odifreddi, P. G. }}; page 151</ref> | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{cite book | title=Classical Recursion Theory, Volume 1| author=Odifreddi, P. G. | publisher=Elsevier| year=1992 | isbn=0-444-89483-7 | pages=668 }} | |||
*{{ cite book | title=Theory of Recursive Functions and Effective Computability | author=Rogers Jr., Hartley | publisher=MIT Press|isbn=0-262-68052-1 | pages=482 | year=1987}} | |||
[[Category:Computability theory]] | |||
Revision as of 11:32, 18 November 2013
In the field of recursion theory, index sets describe classes of partial recursive functions, specifically they give all indices of functions in that class according to a fixed enumeration of partial recursive functions (a Gödel numbering).
Definition
Fix an enumeration of all partial recursive functions, or equivalently of recursively enumerable sets whereby the eth such set is and the eth such function (whose domain is ) is .
Let be a class of partial recursive functions. If then is the index set of . In general is an index set if for every with (i.e. they index the same function), we have . Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index.
Index sets and Rice's theorem
Most index sets are incomputable (non-recursive) aside from two trivial exceptions. This is stated in Rice's theorem:
Let be a class of partial recursive functions with index set . Then is recursive if and only if is empty, or is all of .
where is the set of natural numbers, including zero.
Rice's theorem says "any nontrivial property of partial recursive functions is undecidable"[1]
Notes
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References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534; page 151