Document Open Access Logo

Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton

Author Günter Rote

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2025.86.pdf
  • Filesize: 0.82 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Probabilistic finite automaton
  • Undecidability
  • Post Correspondence Problem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We construct a probabilistic finite automaton (PFA) with 7 states and an input alphabet of 5 symbols for which the PFA Emptiness Problem is undecidable. The only input for the decision problem is the starting distribution. For the proof, we use reductions from special instances of the Post Correspondence Problem.
We also consider some variations: The input alphabet of the PFA can be restricted to a binary alphabet at the expense of a larger number of states. If we allow a rational output value for each state instead of a yes-no acceptance decision, the number of states can even be reduced to 6.

Cite As Get BibTex

Günter Rote. Probabilistic Finite Automaton Emptiness Is Undecidable for a Fixed Automaton. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 86:1-86:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.86

Author Details

Günter Rote
  • Institut für Informatik, Freie Universität Berlin, Germany

References

  1. Vincent D. Blondel and Vincent Canterini. Undecidable problems for probabilistic automata of fixed dimension. Theory Comput. Systems, 36:231-245, 2003. URL: https://doi.org/10.1007/s00224-003-1061-2.
  2. Vincent D. Blondel and John N. Tsitsiklis. The boundedness of all products of a pair of matrices is undecidable. Systems & Control Letters, 41(2):135-140, 2000. URL: https://doi.org/10.1016/S0167-6911(00)00049-9.
  3. Vuong Bui. Growth of Bilinear Maps. PhD thesis, Freie Universität Berlin, Institut für Informatik, 2023. URL: https://doi.org/10.17169/refubium-41705.
  4. Volker Claus. Some remarks on PCP(k) and related problems. Bull. Europ. Assoc. Theoret. Computer Sci. (EATCS), 12:54-61, October 1980. URL: http://page.mi.fu-berlin.de/rote/Kram/Volker_Claus-Some_remarks_on_PCP(k)_and_related_problems-1980-Bull-EATCS-12.pdf.
  5. Volker Claus. The (n, k)-bounded emptiness-problem for probabilistic acceptors and related problems. Acta Informatica, 16:139-160, 1981. URL: https://doi.org/10.1007/BF00261257.
  6. A. Condon and R. J. Lipton. On the complexity of space bounded interactive proofs. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, SFCS '89, pages 462-467, USA, 1989. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1989.63519.
  7. Ruiwen Dong. Recent advances in algorithmic problems for semigroups. ACM SIGLOG News, 10(4):3-23, December 2023. URL: https://doi.org/10.1145/3636362.3636365.
  8. Nathanaël Fijalkow. Undecidability results for probabilistic automata. ACM SIGLOG News, 4(4):10-17, November 2017. URL: https://doi.org/10.1145/3157831.3157833.
  9. Rūsiņš Freivalds. Probabilistic two-way machines. In Jozef Gruska and Michal Chytil, editors, Mathematical Foundations of Computer Science 1981 (MFCS), volume 118 of Lecture Notes in Computer Science, pages 33-45. Springer, 1981. URL: https://doi.org/10.1007/3-540-10856-4_72.
  10. Hugo Gimbert and Youssouf Oualhadj. Probabilistic automata on finite words: Decidable and undecidable problems. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors, Automata, Languages and Programming. 37th International Colloquium, ICALP 2010, Bordeaux, France, July 2010, Proceedings, Part II, volume 6199 of Lecture Notes in Computer Science, pages 527-538, Berlin, Heidelberg, 2010. Springer-Verlag. Full version in https://hal.science/hal-00456538v3. URL: https://doi.org/10.1007/978-3-642-14162-1_44.
  11. Vesa Halava and Tero Harju. On Markov’s undecidability theorem for integer matrices. Semigroup Forum, 75:173-180, 2007. URL: https://doi.org/10.1007/s00233-007-0714-x.
  12. Vesa Halava, Tero Harju, and Mika Hirvensalo. Undecidability bounds for integer matrices using Claus instances. International Journal of Foundations of Computer Science, 18(05):931-948, 2007. URL: https://doi.org/10.1142/S0129054107005066.
  13. Mika Hirvensalo. Improved undecidability results on the emptiness problem of probabilistic and quantum cut-point languages. In Jan van Leeuwen, Giuseppe F. Italiano, Wiebe van der Hoek, Christoph Meinel, Harald Sack, and František Plášil, editors, SOFSEM 2007: Theory and Practice of Computer Science, volume 4362 of Lecture Notes in Computer Science, pages 309-319, Berlin, Heidelberg, 2007. Springer. URL: https://doi.org/10.1007/978-3-540-69507-3_25.
  14. Omid Madani, Steve Hanks, and Anne Condon. On the undecidability of probabilistic planning and related stochastic optimization problems. Artif. Intell., 147(1-2):5-34, 2003. URL: https://doi.org/10.1016/S0004-3702(02)00378-8.
  15. A. Markov. On certain insoluble problems concerning matrices. Doklady Akad. Nauk SSSR (N.S.), 57:539-542, 1947. (Russian). Google Scholar
  16. Yuri Matiyasevich and Géraud Sénizergues. Decision problems for semi-Thue systems with a few rules. Theoretical Computer Science, 330(1):145-169, 2005. URL: https://doi.org/10.1016/j.tcs.2004.09.016.
  17. Masakazu Nasu and Namio Honda. Mappings induced by PGSM-mappings and some recursively unsolvable problems of finite probabilistic automata. Information and Control, 15(3):250-273, 1969. URL: https://doi.org/10.1016/S0019-9958(69)90449-5.
  18. Turlough Neary. Undecidability in binary tag systems and the Post correspondence problem for five pairs of words. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 649-661, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.649.
  19. Michael S. Paterson. Unsolvability in 3× 3 matrices. Stud. in Appl. Math., 49(1):105-107, 1970. URL: https://doi.org/10.1002/sapm1970491105.
  20. Azaria Paz. Introduction to Probabilistic Automata. Computer Science and Applied Mathematics. Academic Press, New York, 1971. URL: https://doi.org/10.1016/c2013-0-11297-4.
  21. Günter Rote. Probabilistic Finite Automaton Emptiness is undecidable, June 2024. URL: https://doi.org/10.48550/arXiv.2405.03035.
  22. Günter Rote. Probabilistic Finite Automaton Emptiness is undecidable for a fixed automaton, December 2024. URL: https://doi.org/10.48550/arXiv.2412.05198.
  23. Paavo Turakainen. Generalized automata and stochastic languages. Proc. Amer. Math. Soc., 21:303-309, 1969. URL: https://doi.org/10.1090/S0002-9939-1969-0242596-1.
  24. Paavo Turakainen. Word-functions of stochastic and pseudo stochastic automata. Annales Fennici Mathematici, 1(1):27-37, February 1975. URL: https://doi.org/10.5186/aasfm.1975.0126.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact