# Sequentiality of String-to-Context Transducers

## TL;DR

This paper extends the characterization of sequential transducers to string-to-context transducers, providing multiple criteria for their sequentiality and analyzing the computational complexity of the equivalence problem.

## Contribution

It introduces the class of string-to-context transducers, extends Choffrut's results, and provides new characterizations and complexity results for their sequentiality.

## Key findings

- Characterization based on Lipschitz property
- Introduction of a new twinning property
- Deciding sequentiality is in coNP

## Abstract

Transducers extend finite state automata with outputs, and describe transformations from strings to strings. Sequential transducers, which have a deterministic behaviour regarding their input, are of particular interest. However, unlike finite-state automata, not every transducer can be made sequential. The seminal work of Choffrut allows to characterise, amongst the functional one-way transducers, the ones that admit an equivalent sequential transducer. In this work, we extend the results of Choffrut to the class of transducers that produce their output string by adding simultaneously, at each transition, a string on the left and a string on the right of the string produced so far. We call them the string-to-context transducers. We obtain a multiple characterisation of the functional string-to-context transducers admitting an equivalent sequential one, based on a Lipschitz property of the function realised by the transducer, and on a pattern (a new twinning property). Last, we prove that given a string-to-context transducer, determining whether there exists an equivalent sequential one is in coNP.