Splitting theorems in recursion theory

Abstract

A splitting of an r.e. set A is a pair A 1, A 2 of disjoint r.e. sets such that A 1 ∪ A 2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, ε, of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees (since A 1 ⩽ T A, A 2 ⩽ T A and A ⩽ T A 1 ⊕ A 2 ). Thus splitting theor ems have been used to obtain results about the structure of ε, the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory.