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).

DefinitionEdit

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  .

where   is the set of natural numbers, including zero.

Rice's theorem says "any nontrivial property of partial recursive functions is undecidable"[1]

NotesEdit

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

ReferencesEdit

  • Odifreddi, P. G. (1992). Classical Recursion Theory, Volume 1. Elsevier. p. 668. ISBN 0-444-89483-7.
  • Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. p. 482. ISBN 0-262-68052-1.