Document Open Access Logo

Synchronizing Deterministic Push-Down Automata Can Be Really Hard

Authors Henning Fernau , Petra Wolf , Tomoyuki Yamakami

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2020.33.pdf
  • Filesize: 0.57 MB
  • 15 pages

Document Identifiers

Author Details

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

Funding

  • Wolf, Petra: DFG project FE 560/9-1

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.MFCS.2020.33

Abstract

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
Keywords
  • Synchronizing automaton
  • Reset sequence
  • Real-time deterministic push-down automaton
  • Finite-turn push-down automaton
  • Computability
  • Computational complexity

Metrics

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

Related Versions

References

  1. Journal of Automata, Languages and Combinatorics - Essays on the Černý Conjecture. https://www.jalc.de/issues/2019/issue_24_2-4/content.html. Accessed: 10 2020.
  2. Rajeev Alur and P. Madhusudan. Visibly Pushdown Languages. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 202-211. ACM, 2004. Google Scholar
  3. Marcelo Arenas, Pablo Barceló, and Leonid Libkin. Regular Languages of Nested Words: Fixed Points, Automata, and Synchronization. Theory of Computing Systems, 49(3):639-670, 2011. Google Scholar
  4. Parvaneh Babari, Karin Quaas, and Mahsa Shirmohammadi. Synchronizing Data Words for Register Automata. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 15:1-15:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  5. Marie-Pierre Béal and Dominique Perrin. Synchronised Automata, page 213–240. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2016. Google Scholar
  6. Jean Berstel. Transductions and Context-Free Languages, volume 38 of Teubner Studienbücher: Informatik. Teubner, 1979. Google Scholar
  7. Meera Blattner. The unsolvability of the equality problem for sentential forms of context-free grammars. J. Comput. Syst. Sci., 7(5):463-468, 1973. URL: https://doi.org/10.1016/S0022-0000(73)80002-9.
  8. Luc Boasson and Géraud Sénizergues. NTS Languages Are Deterministic and Congruential. Journal of Computer and System Sciences, 31(3):332-342, 1985. Google Scholar
  9. Stanislav Böhm and Stefan Göller. Language Equivalence of Deterministic Real-Time One-Counter Automata Is NL-Complete. In Filip Murlak and Piotr Sankowski, editors, Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, volume 6907 of Lecture Notes in Computer Science, pages 194-205. Springer, 2011. Google Scholar
  10. Didier Caucal. Synchronization of Pushdown Automata. In Oscar H. Ibarra and Zhe Dang, editors, Developments in Language Theory, 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006, Proceedings, volume 4036 of Lecture Notes in Computer Science, pages 120-132. Springer, 2006. Google Scholar
  11. Ján Černý. Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikalny Časopis Slovensk, 14(3):208-215, 1964. Google Scholar
  12. Ján Cerný. A Note on Homogeneous Experiments with Finite Automata. Journal of Automata, Languages and Combinatorics, 24(2-4):123-132, 2019. Google Scholar
  13. Dmitry Chistikov, Pavel Martyugin, and Mahsa Shirmohammadi. Synchronizing Automata over Nested Words. Journal of Automata, Languages and Combinatorics, 24(2-4):219-251, 2019. Google Scholar
  14. Wojciech Czerwiǹski, Slawomir Lasota, Ranko Lazić, Jérôme Leroux, and Filip Mazowiecki. The Reachability Problem for Petri Nets is Not Elementary. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 24-33. ACM, 2019. Google Scholar
  15. Laurent Doyen, Line Juhl, Kim Guldstrand Larsen, Nicolas Markey, and Mahsa Shirmohammadi. Synchronizing Words for Weighted and Timed Automata. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, December 15-17, 2014, New Delhi, India, pages 121-132, 2014. Google Scholar
  16. David Eppstein. Reset Sequences for Monotonic Automata. SIAM Journal on Computing, 19(3):500-510, 1990. Google Scholar
  17. Henning Fernau and Petra Wolf. Synchronization of Deterministic Visibly Push-Down Automata. CoRR, abs/2005.01374, 2020. URL: http://arxiv.org/abs/2005.01374.
  18. Henning Fernau, Petra Wolf, and Tomoyuki Yamakami. Synchronizing Deterministic Push-Down Automata Can Be Really Hard. CoRR, abs/2005.01381, 2020. URL: http://arxiv.org/abs/2005.01381.
  19. Emily P. Friedman. The Inclusion Problem for Simple Languages. Theoretical Computer Science, 1(4):297-316, 1976. Google Scholar
  20. Seymour Ginsburg. The Mathematical Theory of Context-Free Languages. McGraw-Hill, 1966. Google Scholar
  21. Seymour Ginsburg and Edwin H Spanier. Finite-Turn Pushdown Automata. SIAM Journal on Control, 4(3):429-453, 1966. Google Scholar
  22. Sheila A. Greibach. Remarks on Blind and Partially Blind One-Way Multicounter Machines. Theoretical Computer Science, 7:311-324, 1978. Google Scholar
  23. Timothy V. Griffiths. The unsolvability of the equivalence problem for lambda-free nondeterministic generalized machines. J. ACM, 15(3):409-413, 1968. URL: https://doi.org/10.1145/321466.321473.
  24. Eitan M. Gurari and Oscar H. Ibarra. The Complexity of Decision Problems for Finite-Turn Multicounter Machines. Journal of Computer and System Sciences, 22(2):220-229, 1981. Google Scholar
  25. Ken Higuchi, Mitsuo Wakatsuki, and Etsuji Tomita. A Polynomial-Time Algorithm for Checking the Inclusion for Real-Time Deterministic Restricted One-Counter Automata Which Accept by Final State. IEICE Transactions on Information and Systems, 78-D(8):939-950, 1995. Google Scholar
  26. Oscar H. Ibarra. The unsolvability of the equivalence problem for epsilon-free ngsm’s with unary input (output) alphabet and applications. SIAM J. Comput., 7(4):524-532, 1978. URL: https://doi.org/10.1137/0207042.
  27. Balázs Imreh and Magnus Steinby. Directable Nondeterministic Automata. Acta Cybernetica, 14(1):105-115, 1999. Google Scholar
  28. Szabolcs Iván. Synchronizing weighted automata. In Zoltán Ésik and Zoltán Fülöp, editors, Proceedings 14th International Conference on Automata and Formal Languages, AFL 2014, Szeged, Hungary, May 27-29, 2014, volume 151 of EPTCS, pages 301-313, 2014. URL: https://doi.org/10.4204/EPTCS.151.21.
  29. Changwook Kim. Quasi-Rocking Real-Time Pushdown Automata. Theoretical Computer Science, 412(48):6720-6735, 2011. Google Scholar
  30. Zvi Kohavi and Niraj K. Jha. Switching and Finite Automata Theory. Cambridge University Press, 3rd edition, 2009. Google Scholar
  31. S. Rao Kosaraju. Decidability of Reachability in Vector Addition Systems (Preliminary Version). In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H. Landweber, editors, Proceedings of the 14th Annual ACM Symposium on Theory of Computing, May 5-7, 1982, San Francisco, California, USA, pages 267-281. ACM, 1982. Google Scholar
  32. Pavel Martyugin. Computational Complexity of Certain Problems Related to Carefully Synchronizing Words for Partial Automata and Directing Words for Nondeterministic Automata. Theory of Computing Systems, 54(2):293-304, 2014. Google Scholar
  33. Yuri V. Matiyasevich and Géraud Sénizergues. Decision Problems for Semi-Thue Systems with a few Rules. Theoretical Computer Science, 330(1):145-169, 2005. Google Scholar
  34. Ernst W. Mayr. An Algorithm for the General Petri Net Reachability Problem. In Proceedings of the 13th Annual ACM Symposium on Theory of Computing, May 11-13, 1981, Milwaukee, Wisconsin, USA, pages 238-246. ACM, 1981. Google Scholar
  35. Kurt Mehlhorn. Pebbling Moutain Ranges and its Application of DCFL-Recognition. In J. W. de Bakker and Jan van Leeuwen, editors, Automata, Languages and Programming, 7th Colloquium, Noordweijkerhout, The Netherlands, July 14-18, 1980, Proceedings, volume 85 of Lecture Notes in Computer Science, pages 422-435. Springer, 1980. Google Scholar
  36. Eitatsu Mikami and Tomoyuki Yamakami. Synchronizing Pushdown Automata and Reset Words, 2020. An article appeared in Japanese as Technical Report of the Institute of Electronics, Information and Communication Engineers, COMP2019-54(2020-03), pp. 57-–63. An English translation is in preparation. Google Scholar
  37. Marvin L Minsky. Recursive Unsolvability of Post’s Problem of "Tag" and Other Topics in Theory of Turing Machines. Annals of Mathematics, pages 437-455, 1961. Google Scholar
  38. 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, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 649-661. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. Google Scholar
  39. Emil L. Post. A Variant of a Recursively Unsolvable Problem. Bulletin of the American Mathematical Society, 52(4):264-268, 1946. Google Scholar
  40. I. K. Rystsov. On Minimizing the Length of Synchronizing Words for Finite Automata. In Theory of Designing of Computing Systems, pages 75-82. Institute of Cybernetics of Ukrainian Acad. Sci., 1980. (in Russian). Google Scholar
  41. I. K. Rystsov. Polynomial Complete Problems in Automata Theory. Information Processing Letters, 16(3):147-151, 1983. Google Scholar
  42. Sven Sandberg. Homing and Synchronizing Sequences. In Manfred Broy, Bengt Jonsson, Joost-Pieter Katoen, Martin Leucker, and Alexander Pretschner, editors, Model-Based Testing of Reactive Systems, Advanced Lectures, volume 3472 of LNCS, pages 5-33. Springer, 2005. Google Scholar
  43. Géraud Sénizergues. The Equivalence and Inclusion Problems for NTS Languages. Journal of Computer and System Sciences, 31(3):303-331, 1985. Google Scholar
  44. Mahsa Shirmohammadi. Qualitative Analysis of Synchronizing Probabilistic Systems. (Analyse qualitative des systèmes probabilistes synchronisants). PhD thesis, École normale supérieure de Cachan, France, 2014. URL: https://tel.archives-ouvertes.fr/tel-01153942.
  45. Yaroslav Shitov. An Improvement to a Recent Upper Bound for Synchronizing Words of Finite Automata. Journal of Automata, Languages and Combinatorics, 24(2-4):367-373, 2019. Google Scholar
  46. Peter H. Starke. Eine Bemerkung über homogene Experimente. Elektronische Informationsverarbeitung und Kybernetik, 2(4):257-259, 1966. Google Scholar
  47. Peter H. Starke. A Remark About Homogeneous Experiments. Journal of Automata, Languages and Combinatorics, 24(2-4):133-137, 2019. Google Scholar
  48. Marek Szykuła. Improving the Upper Bound on the Length of the Shortest Reset Word. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 56:1-56:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  49. Leslie G. Valiant. Decision Procedures for Families of Deterministic Pushdown Automata. PhD thesis, University of Warwick, Coventry, UK, 1973. URL: http://wrap.warwick.ac.uk/34701/.
  50. Mikhail V. Volkov. Synchronizing Automata and the Černý Conjecture. In Carlos Martín-Vide, Friedrich Otto, and Henning Fernau, editors, Language and Automata Theory and Applications, Second International Conference, LATA, volume 5196 of LNCS, pages 11-27. Springer, 2008. Google Scholar
  51. Mitsuo Wakatsuki and Etsuji Tomita. A Fast Algorithm for Checking the Inclusion for Very Simple Deterministic Pushdown Automata. IEICE Transactions on Information and Systems, 76-D(10):1224-1233, 1993. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact