Index set (recursion theory)

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

  1. Odifreddi, P. G. Classical Recursion Theory, Volume 1.; page 151

References

This article is issued from Wikipedia - version of the 2/14/2011. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.