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The Complexity of Reachability Problems in Strongly Connected Finite Automata

Authors Stefan Kiefer , Andrew Ryzhikov

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Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • unambiguous automata
  • nonnegative matrices
  • irreducible matrix sets
  • strongly connected automata
  • matrix monoids
  • mortality
  • completeness
  • synchronisation
  • ergodicity

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Abstract

Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. In particular, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.

Cite As Get BibTex

Stefan Kiefer and Andrew Ryzhikov. The Complexity of Reachability Problems in Strongly Connected Finite Automata. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.62

Author Details

Stefan Kiefer
  • Department of Computer Science, University of Oxford, UK
Andrew Ryzhikov
  • University of Warsaw, Poland

Funding

  • Ryzhikov, Andrew: Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00230.

Acknowledgements

We thank the anonymous reviewers for their comments which improved the presentation of the paper.

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