Ping-pong lemma

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 ${\varphi }_e$.

Let $𝒜$ be a class of partial recursive functions. If $A=\{x:{\varphi }_x\in 𝒜\}$ then $A$ is the index set of $𝒜$. 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 incomputable (non-recursive) aside from two trivial exceptions. This is stated in Rice's theorem:

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

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

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

Notes

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534