  We investigate two problems for a class C of regular word languages. The
C-membership problem asks for an algorithm to decide whether an input language
belongs to C. The C-separation problem asks for an algorithm that, given as
input two regular languages, decides whether there exists a third language in C
containing the first language, while being disjoint from the second. These
problems are considered as means to obtain a deep understanding of the class C.
  It is usual for such classes to be defined by logical formalisms. Logics are
often built on top of each other, by adding new predicates. A natural
construction is to enrich a logic with the successor relation. In this paper,
we obtain simple self-contained proofs of two transfer results: we show that
for suitable logically defined classes, the membership, resp. the separation
problem for a class enriched with the successor relation reduces to the same
problem for the original class.
  Our reductions work both for languages of finite words and infinite words.
The proofs are mostly self-contained, and only require a basic background on
regular languages. This paper therefore gives new, simple proofs of results
that were considered as difficult, such as the decid- ability of the membership
problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy.
