On the complexity of Medvedev reducibility
The Medvedev lattice is a tool used to classify the complexity of effective closed sets. In this article, we are interested in the complexity of using this tool. In particular, we study the complexity of deciding whether two effective closed sets have the same degree, when one is fixed and the other is given as input. This problem is naturally~$Σ^0_3$, and among other results, we show that for all degrees except the maximal one, the problem reaches its maximal complexity. It turns out that the argument is not uniform and depends on the self-similarity of the fixed degree, a notion that we introduce in this article. The complexity of having maximal degree is particularly difficult to establish, but we obtain a lower bound that almost matches the upper bound. We also show that deciding whether an effective closed set has a degree incomparable with a fixed degree $d$ is $Π^0_3$-complete whenever $\mathbf{0}_s <_s d <_s \mathbf{1}_s$.