Relativized depth

Abstract

Bennett's notion of depth is usually considered to describe 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 the notion of depth for such sets, and we investigate for various kinds of oracles whether and how the unrelativized and the relativized version of depth differ. Intuitively speaking, access to an oracle increases computation power. Accordingly, for most notions for sets considered in computability theory, for the corresponding classes trivially for all oracles the unrelativized class is contained in the relativized class or for all oracles the relativized class is contained in the unrelativized class. Examples for these two cases are given by the classes of computable and of Martin-L\"{o}f random sets, respectively. However, in the case for depth the situation is different. It turns out that the classes of deep sets and of sets that are deep relative to the halting set $\emptyset '$ 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-L\"{o}f-random oracle. The set built in the proof of the latter result can also be used to give a short proof of the known fact that every PA-complete degree is Turing-equivalent to the join of two Martin-L\"{o}f-random sets. In fact, we slightly strengthen this result by showing that every DNC$_2$ function is truth-table-equivalent to the join of two Martin-L\"{o}f 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. Obviously, the former case applies to computable oracles. We leave it as an open problem which of the two possibilities can occur for noncomputable K-trivial oracles.