More generally, a subset is recursive if its characteristic function is recursive.
A recursive set is also known as a decidable set or a computable set.
Examples of recursive sets are finite subset of , the set itself, the set of even integers, the set of Fibonacci numbers, the set of pairs where divides , 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.
is computable. In other words, there is an algorithm that halts (and returns ) only when an element in is used as an input.
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 . Below are two examples:
Since can be effectively embedded in , so the notion of recursive sets be extended to subsets of .
Since every finite set can be encoded by the natural numbers, we can define a recursive language over to be a subset such that , when encoded by the natural numbers, is a recursive set. Equivalently, is recursive iff there is a Turing machine that decides (accepts and rejects ).
Using the above definition, one can define a recursive predicate or a recursively enumerable predicate according to whether is a recursive or recursively enumerable set respectively.
|Date of creation||2013-03-22 17:34:52|
|Last modified on||2013-03-22 17:34:52|
|Last modified by||CWoo (3771)|
|Defines||recursively enumerable set|
|Defines||recursively enumerable predicate|