On diagonal functions for equivalence relations

Abstract

We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let E be an equivalence relation on the set of natural numbers \(\omega \), having at least two classes. A total function f is a diagonal function for E if for every x, the numbers x and f(x) are not E-equivalent. It is known that in the case of c.e. relations E, the weak precompleteness of E is equivalent to the lack of computable diagonal functions for E. Here we prove that this result fails already for \(\Delta ^0_2\) equivalence relations, starting with the \(\Pi ^{-1}_2\) level. We focus on the Turing degrees of possible diagonal functions. We prove that for any noncomputable c.e. degree \({\textbf{d}}\), there exists a weakly precomplete c.e. equivalence E admitting a \({\textbf{d}}\)-computable diagonal function. We observe that a Turing degree \({\textbf{d}}\) can compute a diagonal function for every \(\Delta ^0_2\) equivalence relation E if and only if \({\textbf{d}}\) computes \({\textbf{0}}'\). On the other hand, every PA degree can compute a diagonal function for an arbitrary c.e. equivalence E. In addition, if \({\textbf{d}}\) computes diagonal functions for all c.e. E, then \({\textbf{d}}\) must be a DNC degree.