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.

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

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

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

Rice's theorem says "any nontrivial property of partial computable functions is undecidable".¹

Index sets provide many examples of sets which are complete at some level of the arithmetical hierarchy. Here, we say a ${\displaystyle \Sigma _{n}}$ set ${\displaystyle A}$ is ${\displaystyle \Sigma _{n}}$-complete if, for every ${\displaystyle \Sigma _{n}}$ set ${\displaystyle B}$, there is an m-reduction from ${\displaystyle B}$ to ${\displaystyle A}$. ${\displaystyle \Pi _{n}}$-completeness is defined similarly. Here are some examples:²

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

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