Continuous Rational Functions Are Deterministic Regular

Authors Olivier Carton , Gaëtan Douéneau-Tabot

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Olivier Carton
  • IRIF, Université Paris Cité, France
Gaëtan Douéneau-Tabot
  • IRIF, Université Paris Cité, France
  • Direction générale de l'armement - Ingénierie de projets, Paris, France

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Olivier Carton and Gaëtan Douéneau-Tabot. Continuous Rational Functions Are Deterministic Regular. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Transducers
  • infinite words
  • rational functions
  • determinization
  • continuity
  • streaming string transducers
  • two-way transducers


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