Deterministic two-way transducers with pebbles (aka pebble transducers) capture the class of polyregular functions, which extend the string-to-string regular functions allowing polynomial growth instead of linear growth. One of the most fundamental operations on functions is composition, and (poly)regular functions can be realized as a composition of several simpler functions. In general, composition of deterministic two-way transducers incur a doubly exponential blow-up in the size of the inputs. A major improvement in this direction comes from the fundamental result of Dartois et al. [10] showing a polynomial construction for the composition of reversible two-way transducers. A precise complexity analysis for existing composition techniques of pebble transducers is missing. But they rely on the classic composition of two-way transducers and inherit the double exponential complexity. To overcome this problem, we introduce reversible pebble transducers. Our main results are efficient uniformization techniques for non-deterministic pebble transducers to reversible ones and efficient composition for reversible pebble transducers.

Deterministic two-way transducers capture the class of regular functions. The efficiency of composing two-way transducers has a direct implication in algorithmic problems related to reactive synthesis, where transformation specifications are converted into equivalent transducers. These specifications are presented in a modular way, and composing the resultant machines simulates the full specification. An important result by Dartois et al. shows that composition of two-way transducers enjoy a polynomial composition when the underlying transducer is reversible, that is, if they are both deterministic and co-deterministic. This is a major improvement over general deterministic two-way transducers, for which composition causes a doubly exponential blow-up in the size of the inputs in general. Moreover, they show that reversible two-way transducers have the same expressiveness as deterministic two-way transducers. However, the question of expressiveness of reversible transducers over infinite words is still open. In this article, we introduce the class of reversible two-way transducers over infinite words and show that they enjoy the same expressive power as deterministic two-way transducers over infinite words. This is done through a non-trivial, effective construction inducing a single exponential blow-up in the set of states. Further, we also prove that composing two reversible two-way transducers over infinite words incurs only a polynomial complexity, thereby providing foundations for efficient procedure for composition of transducers over infinite words.