Discoveries / theoretical-computer-science

Odifreddi's Problem 3 on Irreducible m-Degrees

Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable $tt$-degree contains a c.e. irreducible $m$-degree, meaning an $m$-degree consisting of a single $1$-degree. Answered negatively: there is a c.e. $tt$-degree containing no c.e. irreducible $m$-degree. This also shows Jockusch's 1969 theorem, which produces an irreducible $m$-degree inside every c.e. $tt$-degree, is strictly optimal and cannot be strengthened to make that degree c.e.