  Tree transductions are binary relations of finite trees. For tree
transductions defined by non-deterministic top-down tree transducers,
inclusion, equivalence and synthesis problems are known to be undecidable.
Adding origin semantics to tree transductions, i.e., tagging each output node
with the input node it originates from, is a known way to recover decidability
for inclusion and equivalence. The origin semantics is rather rigid, in this
work, we introduce a similarity measure for transducers with origin semantics
and show that we can decide inclusion, equivalence and synthesis problems for
origin-close non-deterministic top-down tree transducers.
