Pebble Transducers with Unary Output.

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, which is false with non-unary outputs. Then, we characterize 1-pebble transducers that describe a function computable by a blind transducer, and show that the membership problem is decidable. Our results can be interpreted in terms of automated simplification for iterative programs.