Index set (computability)

Short description: Classes of partial recursive functions

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 [math]\displaystyle{ \varphi_e }[/math] be a computable enumeration of all partial computable functions, and [math]\displaystyle{ W_{e} }[/math] be a computable enumeration of all c.e. sets.

Let [math]\displaystyle{ \mathcal{A} }[/math] be a class of partial computable functions. If [math]\displaystyle{ A = \{x \,:\, \varphi_{x} \in \mathcal{A} \} }[/math] then [math]\displaystyle{ A }[/math] is the index set of [math]\displaystyle{ \mathcal{A} }[/math]. In general [math]\displaystyle{ A }[/math] is an index set if for every [math]\displaystyle{ x,y \in \mathbb{N} }[/math] with [math]\displaystyle{ \varphi_x \simeq \varphi_y }[/math] (i.e. they index the same function), we have [math]\displaystyle{ x \in A \leftrightarrow y \in A }[/math]. 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 [math]\displaystyle{ \mathcal{C} }[/math] be a class of partial computable functions with its index set [math]\displaystyle{ C }[/math]. Then [math]\displaystyle{ C }[/math] is computable if and only if [math]\displaystyle{ C }[/math] is empty, or [math]\displaystyle{ C }[/math] is all of [math]\displaystyle{ \mathbb{N} }[/math].

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 [math]\displaystyle{ \Sigma_n }[/math] set [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \Sigma_n }[/math]-complete if, for every [math]\displaystyle{ \Sigma_n }[/math] set [math]\displaystyle{ B }[/math], there is an m-reduction from [math]\displaystyle{ B }[/math] to [math]\displaystyle{ A }[/math]. [math]\displaystyle{ \Pi_n }[/math]-completeness is defined similarly. Here are some examples:

• [math]\displaystyle{ \mathrm{Emp} = \{ e \,:\, W_e = \varnothing \} }[/math] is [math]\displaystyle{ \Pi_1 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Fin} = \{ e \,:\, W_e \text{ is finite} \} }[/math] is [math]\displaystyle{ \Sigma_2 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Inf} = \{ e \,:\, W_e \text{ is infinite} \} }[/math] is [math]\displaystyle{ \Pi_2 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Tot} = \{ e \,:\, \varphi_e \text{ is total} \} = \{ e : W_e = \mathbb{N} \} }[/math] is [math]\displaystyle{ \Pi_2 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Con} = \{ e \,:\, \varphi_e \text{ is total and constant} \} }[/math] is [math]\displaystyle{ \Pi_2 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Cof} = \{ e \,:\, W_e \text{ is cofinite} \} }[/math] is [math]\displaystyle{ \Sigma_3 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Rec} = \{ e \,:\, W_e \text{ is computable} \} }[/math] is [math]\displaystyle{ \Sigma_3 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Ext} = \{ e \,:\, \varphi_e \text{ is extendible to a total computable function} \} }[/math] is [math]\displaystyle{ \Sigma_3 }[/math]-complete.
• [math]\displaystyle{ \mathrm{Cpl} = \{ e \,:\, W_e \equiv_\mathrm{T} \mathrm{HP} \} }[/math] is [math]\displaystyle{ \Sigma_4 }[/math]-complete, where [math]\displaystyle{ \mathrm{HP} }[/math] is the halting problem.

Empirically, if the "most obvious" definition of a set [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \Sigma_n }[/math] [resp. [math]\displaystyle{ \Pi_n }[/math]], we can usually show that [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \Sigma_n }[/math]-complete [resp. [math]\displaystyle{ \Pi_n }[/math]-complete].

References

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