Continuous rational functions are deterministic regular

TL;DR

This paper characterizes the class of infinite-word rational functions that are computable by deterministic two-way transducers, showing they are exactly the continuous rational functions, and provides a decision procedure for this property.

Contribution

It identifies the class of continuous rational functions over infinite words as precisely those computable by deterministic two-way transducers, solving an open problem.

Findings

Continuous rational functions over infinite words are exactly those computable by deterministic two-way transducers.

The property of continuity for rational functions over infinite words is decidable.

The paper resolves an open question from previous research on infinite-word transducers.

Abstract

A word-to-word function is rational if it can be realized by a non-deterministic one-way transducer. Over finite words, it is a classical result that any rational function is regular, i.e. it can be computed by a deterministic two-way transducer, or equivalently, by a deterministic streaming string transducer (a one-way automaton which manipulates string registers). This result no longer holds for infinite words, since a non-deterministic one-way transducer can guess, and check along its run, properties such as infinitely many occurrences of some pattern, which is impossible for a deterministic machine. In this paper, we identify the class of rational functions over infinite words which are also computable by a deterministic two-way transducer. It coincides with the class of rational functions which are continuous, and this property can thus be decided. This solves an open question raised in a previous paper of Dave et al.