Computational Complexity of Synchronization under Regular Constraints

Authors Henning Fernau , Vladimir V. Gusev , Stefan Hoffmann , Markus Holzer , Mikhail V. Volkov , Petra Wolf

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Author Details

Henning Fernau
  • Fachbereich 4 - Abteilung Informatikwissenschaften, Universität Trier, Germany
Vladimir V. Gusev
  • Leverhulme Research Centre for Functional Materials Design, University of Liverpool, UK
Stefan Hoffmann
  • Fachbereich 4 - Abteilung Informatikwissenschaften, Universität Trier, Germany
Markus Holzer
  • Institut für Informatik, Universität Gießen, Germany
Mikhail V. Volkov
  • Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, Russia
Petra Wolf
  • Fachbereich 4 - Abteilung Informatikwissenschaften, Universität Trier, Germany


This project started during the workshop "Modern Complexity Aspects of Formal Languages" that took place at Trier University on February 11-15, 2019.

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Henning Fernau, Vladimir V. Gusev, Stefan Hoffmann, Markus Holzer, Mikhail V. Volkov, and Petra Wolf. Computational Complexity of Synchronization under Regular Constraints. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Problems, reductions and completeness
  • Finite automata
  • synchronization
  • computational complexity


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