Bounded Turing Reductions and Data Processing Inequalities for Sequences

TL;DR

This paper establishes data processing inequalities for sequences under bounded Turing reductions, showing how shared information measures are preserved or can increase under certain computable transformations.

Contribution

It introduces new data processing inequalities for sequences using bounded Turing functionals and mutual dimensions, extending understanding of information preservation and increase.

Findings

Shared information does not increase under Lipschitz reducibility.

Adjustments to Turing functional bounds yield different inequalities.

Reverse inequalities show possible significant increases in shared information.

Abstract

A data processing inequality states that the quantity of shared information between two entities (e.g. signals, strings) cannot be significantly increased when one of the entities is processed by certain kinds of transformations. In this paper, we prove several data processing inequalities for sequences, where the transformations are bounded Turing functionals and the shared information is measured by the lower and upper mutual dimensions between sequences. We show that, for all sequences $X,Y,$ and $Z$, if $Z$ is computable Lipschitz reducible to $X$, then \[ mdim(Z:Y) \leq mdim(X:Y) \text{ and } Mdim(Z:Y) \leq Mdim(X:Y). \] We also show how to derive different data processing inequalities by making adjustments to the computable bounds of the use of a Turing functional. The yield of a Turing functional $\Phi^S$ with access to at most $n$ bits of the oracle $S$ is the smallest input $m \in \mathbb{N}$ such that $\Phi^{S \upharpoonright n}(m)\uparrow$. We show how to derive reverse data processing inequalities (i.e., data processing inequalities where the transformation may significantly increase the shared information between two entities) for sequences by applying computable bounds to the yield of a Turing functional.