Synchronizing Deterministic Push-Down Automata Can Be Really Hard

Authors Henning Fernau , Petra Wolf , Tomoyuki Yamakami

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Henning Fernau
  • Universität Trier, Fachbereich IV, Informatikwissenschaften, Germany
Petra Wolf
  • Universität Trier, Fachbereich IV, Informatikwissenschaften, Germany
Tomoyuki Yamakami
  • University of Fukui, Faculty of Engineering, Japan

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Henning Fernau, Petra Wolf, and Tomoyuki Yamakami. Synchronizing Deterministic Push-Down Automata Can Be Really Hard. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata. There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Automata extensions
  • Theory of computation → Transducers
  • Synchronizing automaton
  • Reset sequence
  • Real-time deterministic push-down automaton
  • Finite-turn push-down automaton
  • Computability
  • Computational complexity


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