Characterization of recursively enumerable sets

Abstract

Let N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N² operations of transposition, composition, and bracketing are defined as follows: P^∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃_{n ∈ M}(Pⁿ Q Rⁿ).

Theorem. The class of recursively enumerable subsets of N² is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.

This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P^∪, A^∪M, R].

The above definitions of the transposition, composition, and bracketing operations on subsets of N² can be generalized to subsets of X² for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.