  Concatenation hierarchies are classifications of regular languages. All such
hierarchies are built through the same construction process: start from an
initial class of languages and build new levels using two generic operations.
Concatenation hierarchies have gathered a lot of interest since the 70s, thanks
to an alternate logical definition: each concatenation hierarchy is the
quantification alternation hierarchy within a variant of first-order logic over
words.
  Our goal is to understand such hierarchies. We look at two decision problems:
membership and separation. For a class of languages C, C-separation takes two
regular languages as input and asks whether there exists a third one in C
including the first one and disjoint from the second one. Settling whether
separation is decidable for the levels within a given concatenation hierarchy
is among the most fundamental and challenging questions in formal language
theory. In all prominent cases, it is open, or answered positively for low
levels only. Recently, a breakthrough was made using a generic approach for a
specific kind of hierarchy: those with a finite basis. In this case, separation
is always decidable for levels 1/2, 1 and 3/2.
  Our main theorem is similar but independent: we consider hierarchies with
possibly infinite bases, but that may only contain group languages. An example
is the quantifier alternation hierarchy of first-order logic with modular
predicates: its basis consists of languages counting the length of words modulo
some number. Using a generic approach, we show that for any such hierarchy, if
separation is decidable for the basis, then it is decidable for levels up to
3/2. This complements the aforementioned result nicely: all bases considered in
the literature are either finite or made of group languages. Thus, one may
handle the lower levels of any prominent hierarchy in a generic way.
