Pebble transducers are nested two-way transducers which can drop marks (named "pebbles") on their input word. Blind transducers have been introduced by Nguyên 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.

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Pebble transducers are nested two-way transducers which can drop marks (named “pebbles”) on their input word. Such machines can compute functions whose output size is polynomial in the size of their input. They can be seen as simple recursive programs whose recursion height is bounded. A natural problem is, given a pebble transducer, to compute an equivalent pebble transducer with minimal recursion height. This problem has been open since the introduction of the model.In this paper, we study two restrictions of pebble transducers, that cannot see the marks (“blind pebble transducers” introduced by Nguyên et al.), or that can only see the last mark dropped (“last pebble transducers” introduced by Engelfriet et al.). For both models, we provide an effective algorithm for minimizing the recursion height. The key property used in both cases is that a function whose output size is linear (resp. quadratic, cubic, etc.) can always be computed by a machine whose recursion height is 1 (resp. 2, 3, etc.). We finally show that this key property fails as soon as we consider machines that can see more than one mark.