Index set (recursion theory)

Fix an enumeration of all partial recursive functions, or equivalently of recursively enumerable sets whereby the eth such set is ${\displaystyle W_{e}}$ and the eth such function (whose domain is ${\displaystyle W_{e}}$) is ${\displaystyle \phi _{e}}$.

Let ${\displaystyle {\mathcal {A}}}$ be a class of partial recursive functions. If ${\displaystyle A=\{x:\phi _{x}\in {\mathcal {A}}\}}$ then ${\displaystyle A}$ is the index set of ${\displaystyle {\mathcal {A}}}$. In general ${\displaystyle A}$ is an index set if for every ${\displaystyle x,y\in \mathbb {N} }$ with ${\displaystyle \phi _{x}\simeq \phi _{y}}$ (i.e. they index the same function), we have ${\displaystyle x\in A\leftrightarrow y\in A}$. Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index.

Most index sets are incomputable (non-recursive) aside from two trivial exceptions. This is stated in Rice's theorem:

Let ${\displaystyle {\mathcal {C}}}$ be a class of partial recursive functions with index set ${\displaystyle C}$. Then ${\displaystyle C}$ is recursive if and only if ${\displaystyle C}$ is empty, or ${\displaystyle C}$ is all of ${\displaystyle \omega }$.

where ${\displaystyle \omega }$ is the set of natural numbers, including zero.

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

• 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.