  The classical decision problem, as it is understood today, is the quest for a
delineation between the decidable and the undecidable parts of first-order
logic based on elegant syntactic criteria. In this paper, we treat the concept
of separateness of variables and explore its applicability to the classical
decision problem. Two disjoint sets of first-order variables are separated in a
given formula if variables from the two sets never co-occur in any atom of that
formula. This simple notion facilitates extending many well-known decidable
first-order fragments significantly and in a way that preserves decidability.
We will demonstrate that for several prefix fragments, several guarded
fragments, the two-variable fragment, and for the fluted fragment. Altogether,
we will investigate nine such extensions more closely. Interestingly, each of
them contains the relational monadic first-order fragment without equality.
Although the extensions exhibit the same expressive power as the respective
originals, certain logical properties can be expressed much more succinctly. In
three cases the succinctness gap cannot be bounded using any elementary
function.
