Recursive and Recursively Enumerable Sets

Recursive and Recursively Enumerable Sets

A set T of integers is said to be recursively enumerable if it constitutes the range of a recursive function, i.e., if there exists a recursive function that can eventually generate any element in T (Wolfram 2002, p. 1138). Any recursive set is...

A set T of integers is said to be recursively enumerable if it constitutes the range of a recursive function, i.e., if there exists a recursive function that can eventually generate any element in T (Wolfram 2002, p. 1138). Any recursive set is also recursively enumerable.

The union and intersection of two recursively enumerable sets are also recursively enumerable.

Recursively undecidable problems give examples of recursively enumerable sets that are not recursive. For example, convergence of phi_x(x) is known to be recursively undecidable, where phi_x denotes the Turing machine with Gödel number x. Hence the set of all x for which phi_x(x) is convergent is not recursive. However, this set is recursively enumerable because it is the range of f defined as follows:

f(x)={x if phi_x(x) is convergent; divergent if phi_x(x) is divergent.
(1) A set A is recursive iff both A and its complement are recursively enumerable. This provides an approach to constructing additional sets that are not recursively enumerable. In particular, the set of all Gödel numbers of total Turing machines is an example of a set which is not recursively enumerable.

The complements of recursively enumerable but not recursive sets are all not recursively enumerable, although complements of sets that are not recursively enumerable are not necessarily recursively enumerable. For instance, the complement of the set of all Gödel numbers of total Turing machines is not recursively enumerable.

One of fundamental properties of recursively enumerable sets is that they could be alternatively defined by domains as opposed to ranges. In particular, a set A is recursively enumerable iff A is the domain of a recursive function.

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