Relativized depth

TL;DR

This paper explores how the concept of depth, related to the usefulness and structure of information in sets, changes when relativized to different oracles, revealing complex relationships and new insights in computability theory.

Contribution

It introduces a relativized notion of depth, analyzes its properties across various oracles, and establishes new relationships between deep sets, random sets, and computational degrees.

Findings

Deep sets and their relativized counterparts are incomparable.

Deep sets are strictly contained in sets deep relative to any Martin-Löf-random oracle.

Every DNC$_2$ function is truth-table-equivalent to the join of two Martin-Löf random sets.

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.