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).
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 theoremEdit
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 .
Rice's theorem says "any nontrivial property of partial recursive functions is undecidable"
- Odifreddi, P. G. Classical Recursion Theory, Volume 1.; page 151