  We introduce a hierarchy of degree structures between the Medvedev and
Muchnik lattices which allow varying amounts of non-uniformity. We use these
structures to introduce the notion of the uniformity of a Muchnik reduction,
which expresses how uniform a reduction is. We study this notion for several
well-known reductions from algorithmic randomness. Furthermore, since our new
structures are Brouwer algebras, we study their propositional theories.
Finally, we study if our new structures are elementarily equivalent to each
other.

Logic