recursive set

A subset $S$ of the natural numbers $\mathbb{N}$ is said to be recursive if its characteristic function

${\chi }_S(x)$ $:=$

is recursive (computable). In other words, there is an algorithm (via Turing machine for example) that determines whether an element is in $S$ or not in $S$.

More generally, a subset $S\subseteq {\mathbb{N}}^n$ is recursive if its characteristic function $f_S$ is recursive.

A recursive set is also known as a decidable set or a computable set.

Examples of recursive sets are finite subset of $\mathbb{N}$, the set $\mathbb{N}$ itself, the set of even integers, the set of Fibonacci numbers, the set of pairs $(a,b)$ where $a$ divides $b$, and the set of prime numbers. In the last example, one may use the Sieve of Eratosthenes as an algorithm to determine the primality of an integer.

A set $S\subseteq \mathbb{N}$ is recursively enumerable if the partial function

$f(x)$ $:=$

is computable. In other words, there is an algorithm that halts (and returns $1$) only when an element in $S$ is used as an input.

Remarks

• A special case of a recursive set is that of a primitive recursive set. A set is primitive recursive if its characteristic function is primitive recursive (http://planetmath.org/PrimitiveRecursive). All of the examples cited above are primitive recursive.

• On the other hand, one can broaden the scope of recursiveness to sets which are not necessarily subsets of ${\mathbb{N}}^n$. Below are two examples:

  • Since $\mathbb{Z}$ can be effectively embedded in $\mathbb{N}$, so the notion of recursive sets be extended to subsets of $\mathbb{Z}$.

  • Since every finite set $\mathrm{\Sigma }$ can be encoded by the natural numbers, we can define a recursive language over $\mathrm{\Sigma }$ to be a subset $L\subseteq {\mathrm{\Sigma }}^{*}$ such that $L$, when encoded by the natural numbers, is a recursive set. Equivalently, $L$ is recursive iff there is a Turing machine that decides $L$ (accepts $L$ and rejects ${\mathrm{\Sigma }}^{*}-L$).

• Similarly, recursive enumerability can be defined on languages: a language $L$ over $\mathrm{\Sigma }$ is recursively enumerable if its encoding by the natural numbers is a recursively enumerable set. This is equivalent to saying that $L$ is accepted by a Turing machine.

• Using the above definition, one can define a recursive predicate or a recursively enumerable predicate $\phi (x)$ according to whether $\{x\mid \phi (x)\}$ is a recursive or recursively enumerable set respectively.

Title	recursive set
Canonical name	RecursiveSet
Date of creation	2013-03-22 17:34:52
Last modified on	2013-03-22 17:34:52
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	16
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03B25
Classification	msc 03D20
Synonym	decidable set
Synonym	computable set
Synonym	decidable predicate
Synonym	computable predicate
Defines	recursively enumerable set
Defines	recursive predicate
Defines	recursively enumerable predicate
Defines	recursive language