Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if:

Or, equivalently,

The first condition suggests why the term semidecidable is sometimes used; the second suggests why computably enumerable is used. The abbreviations r.e. and c.e. are often used, even in print, instead of the full phrase.

In computational complexity theory, the complexity class containing all recursively enumerable sets is RE. In recursion theory, the lattice of r.e. sets under inclusion is denoted \mathcal{E}.

Contents

Formal definition

A set S of natural numbers is called recursively enumerable if there is a partial recursive function (synonymously, a partial computable function) whose domain is exactly S, meaning that the function is defined if and only if its input is a member of S.

Equivalent formulations

The following are all equivalent properties of a set S of natural numbers:

Semidecidability:
  • The set S is recursively enumerable. That is, S is the domain (co-range) of a partial recursive function.
  • There is a partial recursive function f such that:
f(x) = \left\{\begin{matrix} 0 &\mbox{if}\ x \in S \\ \mbox{undefined/does not halt}\ &\mbox{if}\ x \notin S \end{matrix}\right.
Enumerability:
  • The set S is the range of a partial recursive function.
  • The set S is the range of a total recursive function or empty. If S is infinite, the function can be chosen to be injective.
  • The set S is the range of a primitive recursive function or empty. Even if S is infinite, repetition of values may be necessary in this case.
Diophantine:
  • There is a polynomial p with integer coefficients and variables x, a, b, c, d, e, f, g, h, i ranging over the natural numbers such that
x \in S \Leftrightarrow \exists a,b,c,d,e,f,g,h,i \ ( p(x,a,b,c,d,e,f,g,h,i) = 0).
  • There is a polynomial from the integers to the integers such that the set S contains exactly the non-negative numbers in its range.

The equivalence of semidecidability and enumerability can be obtained by the technique of dovetailing.

The Diophantine characterizations of a recursively enumerable set, while not as straightforward or intuitive as the first definitions, were found by Yuri Matiyasevich as part of the negative solution to Hilbert's Tenth Problem. Diophantine sets predate recursion theory and are therefore historically the first way to describe these sets (although this equivalence was only remarked more than three decades after the introduction of recursively enumerable sets). The number of bound variables in the above definition of the Diophantine set is the best known so far; it might be that a lower number can be used to define all diophantine sets.

Examples

Properties

If A and B are recursively enumerable sets then A ∩ B, A ∪ B and A × B (with the ordered pair of natural numbers mapped to a single natural number with the Cantor pairing function) are recursively enumerable sets. The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set.

A set is recursively enumerable if and only if it is at level \Sigma^0_1 of the arithmetical hierarchy.

A set T is called co-recursively enumerable or co-r.e. if its complement \mathbb{N} \setminus T is recursively enumerable. Equivalently, a set is co-r.e. if and only if it is at level \Pi^0_1 of the arithmetical hierarchy.

A set A is recursive (synonym: computable) if and only if both A and the complement of A are recursively enumerable. A set is recursive if and only if it is either the range of an increasing total recursive function or finite.

Some pairs of recursively enumerable sets are effectively separable and some are not.

Remarks

According to the Church-Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is recursively enumerable if and only if there is some algorithm which yields an enumeration of S. This cannot be taken as a formal definition, however, because the Church-Turing thesis is an informal conjecture rather than a formal axiom.

The definition of a recursively enumerable set as the domain of a partial function, rather than the range of a total recursive function, is common in contemporary texts. This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory, the definition corresponding to domains has been found to be more natural. Other texts use the definition in terms of enumerations, which is equivalent for recursively enumerable sets.

References