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Milds # Index set (computability)

In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according to a fixed Gödel numbering of partial computable functions.

## Definition

Let $\varphi _{e}$ be a computable enumeration of all partial computable functions, and $W_{e}$ be a computable enumeration of all c.e. sets.

Let ${\mathcal {A}}$ be a class of partial computable functions. If $A=\{x:\varphi _{x}\in {\mathcal {A}}\}$ 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 $\varphi _{x}\simeq \varphi _{y}$ (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 non-computable, aside from two trivial exceptions. This is stated in Rice's theorem:

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

Rice's theorem says "any nontrivial property of partial computable 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].

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