# One-way definability of two-way word transducers

## TL;DR

This paper investigates when two-way word transducers can be equivalently implemented by one-way transducers, providing new algorithms with specific complexity bounds and optimal constructions for certain cases.

## Contribution

It introduces a new algorithm to decide one-way definability of two-way transducers with improved complexity bounds and constructs equivalent one-way transducers when possible.

## Key findings

- Decidability of one-way definability with doubly exponential space complexity
- Optimal construction of equivalent one-way transducers in the sweeping case
- Complexity reduction for sweeping transducers compared to general two-way transducers

## Abstract

Functional transductions realized by two-way transducers (or, equally, by streaming transducers or MSO transductions) are the natural and standard notion of "regular" mappings from words to words. It was shown in 2013 that it is decidable if such a transduction can be implemented by some one-way transducer, but the given algorithm has non-elementary complexity. We provide an algorithm of different flavor solving the above question, that has doubly exponential space complexity. In the special case of sweeping transducers the complexity is one exponential less. We also show how to construct an equivalent one-way transducer, whenever it exists, in doubly or triply exponential time, again depending on whether the input transducer is sweeping or two-way. In the sweeping case our construction is shown to be optimal.