  Boja\'nczyk recently initiated an intensive study of deterministic pebble
transducers, which are two-way automata that can drop marks (named "pebbles")
on their input word, and produce an output word. They describe functions from
words to words. Two natural restrictions of this definition have been
investigated: marble transducers by Dou\'eneau-Tabot et al., and
comparison-free pebble transducers (that we rename here "blind transducers") by
Nguy\^en et al.
  Here, we study the decidability of membership problems between the classes of
functions computed by pebble, marble and blind transducers that produce a unary
output. First, we show that pebble and marble transducers have the same
expressive power when the outputs are unary (which is false over non-unary
outputs). Then, we characterize 1-pebble transducers with unary output that
describe a function computable by a blind transducer, and show that the
membership problem is decidable. These results can be interpreted in terms of
automated simplification of programs.
