  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.
