  In the constrained synchronization problem we ask if a given automaton admits
a synchronizing word coming from a fixed regular constraint language. We show
that intersecting a given constraint language with an ideal language decreases
the computational complexity. Additionally, we state a theorem giving
PSPACE-hardness that broadly generalizes previously used constructions and a
result on how to combine languages by concatenation to get polynomial time
solvable constrained synchronization problems. We use these results to give a
classification of the complexity landscape for small constraint automata of up
to three states.
