Pebble transducers with unary output

TL;DR

This paper investigates the expressive power and decidability of membership problems for classes of functions computed by pebble, marble, and blind transducers with unary output, revealing key equivalences and decidability results.

Contribution

It proves pebble and marble transducers are equally expressive with unary outputs and characterizes 1-pebble transducers that are equivalent to blind transducers, establishing decidability of membership.

Findings

Pebble and marble transducers have the same power with unary outputs.

Membership problem for 1-pebble and blind transducers is decidable.

Results aid automated program simplification.

Abstract

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.