Hiding Pebbles When the Output Alphabet Is Unary

Pebble transducers are nested two-way transducers which can drop marks (named "pebbles") on their input word. Blind transducers have been introduced by Nguy\^en et al. as a subclass of pebble transducers, which can nest two-way transducers but cannot drop pebbles on their input. In this paper, we study the classes of functions computed by pebble and blind transducers, when the output alphabet is unary. We first provide characterizations of these classes in terms of Cauchy and Hadamard products, in the spirit of rational series. Then, we show how to decide if a function computed by a pebble transducer can be computed by a blind transducer. This result also provides a pumping-like characterization of the functions computed by blind transducers.