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.

我们使用 Badaev 引入的弱预完备等价关系。弱预完备性是受可计算理论中各种不动点定理启发的一个自然概念。设E是自然数集\(\omega \)上的等价关系，至少有两个类。如果对于每个x，数字x和f ( x ) 不等价，则总函数f是E的对角函数。众所周知，在 ce 关系E的情况下， E的弱预完备性相当于E缺乏可计算的对角函数。在这里，我们证明这个结果对于\(\Delta ^0_2\)等价关系已经失败，从\(\Pi ^{-1}_2\)级别开始。我们关注可能的对角函数的图灵度。我们证明，对于任何不可计算的 ce 度\({\textbf{d}}\)，存在一个弱预完备 ce 等价E，承认一个\({\textbf{d}}\)可计算的对角函数。我们观察到图灵度\({\textbf{d}}\)可以为每个\(\Delta ^0_2\)等价关系E计算对角函数当且仅当\({\textbf{d}}\)计算\({\textbf{0}}'\)。另一方面，每个 PA 度都可以计算任意 ce 等价E的对角函数。此外，如果\({\textbf{d}}\)计算所有 ce E的对角函数，则\({\textbf{d}}\)必须是 DNC 度。