Relativized depth

Abstract

Bennett's notion of depth is usually considered to describe, roughly speaking, the usefulness and internal organization of the information encoded into an object such as an infinite binary sequence. We consider a natural way to relativize this notion, and we investigate for various kinds of oracles whether and how the unrelativized and the relativized version of depth differ.While most classes emerging from computability theory, once relativized to some oracle, either contain or are contained in their unrelativized version, this is not the case of depth: it turns out that the classes of deep sets and of sets that are deep relative to the halting set ∅′ are incomparable with respect to set-theoretical inclusion.On the other hand, the class of deep sets is strictly contained in the class of sets that are deep relative to any given Martin-Löf-random oracle. The set built to show this separation can also be used to prove that every DNC2 function is truth-table-equivalent to the symmetric difference of two Martin-Löf random sets, giving an alternative proof of the known fact that every PA-complete degree is truth-table equivalent to the join of two Martin-Löf random sets.Furthermore, we observe that the class of deep sets relative to any given K-trivial oracle either is the same as or is strictly contained in the class of deep sets. We leave as an open problem which of the two possibilities can occur for noncomputable K-trivial oracles.