Complexity of Preimage Problems for Deterministic Finite Automata

Authors Mikhail V. Berlinkov, Robert Ferens, Marek Szykula

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Mikhail V. Berlinkov
  • Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg, Russia
Robert Ferens
  • Institute of Computer Science, University of Wrocław, Wrocław, Poland
Marek Szykula
  • Institute of Computer Science, University of Wrocław, Wrocław, Poland

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Mikhail V. Berlinkov, Robert Ferens, and Marek Szykula. Complexity of Preimage Problems for Deterministic Finite Automata. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally consider parametrized complexity by the size of the given subset. We focus on the most interesting cases that are the subclasses of strongly connected, synchronizing, and binary automata.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • avoiding word
  • extending word
  • extensible subset
  • reset word
  • synchronizing automaton


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