Index set (computability)

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 $W_{e}$ and the eth such function (whose domain is $W_{e}$) is $\phi _{e}$.

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

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

Rice's theorem says "any nontrivial property of partial recursive functions is undecidable".

Completeness in the arithmetical hierarchy

Index sets provide many examples of sets which are complete at some level of the arithmetical hierarchy. Here, we say a $\Sigma _{n}$ set $A$ is $\Sigma _{n}$-complete if, for every $\Sigma _{n}$ set $B$, there is an m-reduction from $B$ to $A$. $\Pi _{n}$-completeness is defined similarly. Here are some examples:

• $\mathrm {Emp} =\{e:W_{e}=\varnothing \}$ is $\Pi _{1}$-complete.
• $\mathrm {Fin} =\{e:W_{e}{\text{ is finite}}\}$ is $\Sigma _{2}$-complete.
• $\mathrm {Inf} =\{e:W_{e}{\text{ is infinite}}\}$ is $\Pi _{2}$-complete.
• $\mathrm {Tot} =\{e:\varphi _{e}{\text{ is total}}\}=\{e:W_{e}=\mathbb {N} \}$ is $\Pi _{2}$-complete.
• $\mathrm {Con} =\{e:\varphi _{e}{\text{ is total and constant}}\}$ is $\Pi _{2}$-complete.
• $\mathrm {Cof} =\{e:W_{e}{\text{ is cofinite}}\}$ is $\Sigma _{3}$-complete.
• $\mathrm {Rec} =\{e:W_{e}{\text{ is computable}}\}$ is $\Sigma _{3}$-complete.
• $\mathrm {Ext} =\{e:\varphi _{e}{\text{ is extendible to a total computable function}}\}$ is $\Sigma _{3}$-complete.
• $\mathrm {Cpl} =\{e:W_{e}\equiv _{\mathrm {T} }\mathrm {HP} \}$ is $\Sigma _{4}$-complete, where $\mathrm {HP}$ is the halting problem.

Empirically, if the "most obvious" definition of a set $A$ is $\Sigma _{n}$ [resp. $\Pi _{n}$], we can usually show that $A$ is $\Sigma _{n}$-complete [resp. $\Pi _{n}$-complete].

References

• 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.