Document Open Access Logo

Polynomially Ambiguous Probabilistic Automata on Restricted Languages (Track B: Automata, Logic, Semantics, and Theory of Programming)

Author Paul C. Bell

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.105.pdf
  • Filesize: 465 kB
  • 14 pages

Document Identifiers

Author Details

Paul C. Bell
  • Department of Computer Science, Byrom Street, Liverpool John Moores University, Liverpool, L3-3AF, UK

Acknowledgements

We thank the referees for their careful reading of this manuscript and their helpful improvements.

Cite As Get BibTex

Paul C. Bell. Polynomially Ambiguous Probabilistic Automata on Restricted Languages (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 105:1-105:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.105

Abstract

We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for non-strict cut-points of polynomially ambiguous PFA remains undecidable even when the input word is over a bounded language and all PFA transition matrices are commutative. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freeness/injectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantitative automata
  • Theory of computation → Probabilistic computation
Keywords
  • Probabilistic finite automata
  • ambiguity
  • undecidability
  • bounded language
  • emptiness

Metrics

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

References

  1. L. Babai, R. Beals, J-Y. Cai, G. Ivanyos, and E. M. Luks. Multiplicative equations over commuting matrices. In Proc. of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 96, 1996. Google Scholar
  2. L. Babai and S. Moran. Arthur–Merlin games: a randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36:254-276, 1988. Google Scholar
  3. P. C. Bell, S. Chen, and L. M. Jackson. Scalar ambiguity and freeness in matrix semigroups over bounded languages. In Language and Automata Theory and Applications, volume LNCS 9618, pages 493-505, 2016. Google Scholar
  4. P. C. Bell, V. Halava, and M. Hirvensalo. Decision Problems for Probabilistic Finite Automata on Bounded Languages. Fundamenta Informaticae, 123(1):1-14, 2012. Google Scholar
  5. A. Bertoni. The solution of problems relative to probabilistic automata in the frame of the formal language theory. GI Jahrestagung, pages 107-112, 1974. Google Scholar
  6. A. Bertoni, G. Mauri, and M. Torelli. Some recursively unsolvable problems relating to isolated cutpoints in probabilistic automata. In Automata, Languages and Programming, volume 52, pages 87-94, 1977. Google Scholar
  7. V. Blondel and V. Canterini. Undecidable problems for probabilistic automata of fixed dimension. Theory of Computing Systems, 36:231-245, 2003. Google Scholar
  8. V. Blondel and J. Tsitsiklis. The boundedness of all products of a pair of matrices is undecidable. Systems and Control Letters, Elsevier, 41:2:135-140, 2000. Google Scholar
  9. V. Blondel and J. N. Tsitsiklis. A survey of computational complexity results in systems and control. Automatica, 36:1249-1274, 2000. Google Scholar
  10. R. G. Bukharaev. Probabilistic automata. Journal of Mathematical Sciences, 13(3):359-386, 1980. Google Scholar
  11. J. Buresh-Oppenheim, M. Clegg, R. Impagliazzo, and T. Pitassi. Homogenization and the polynomial calculus. Computational complexity, 11(3-4):91-108, 2002. Google Scholar
  12. J. Cassaigne, J. Karhumäki, and T. Harju. On the Decidability of the Freeness of Matrix Semigroups. International Journal of Algebra and Computation, 9(3-4):295-305, 1999. Google Scholar
  13. É. Charlier and J. Honkala. The freeness problem over matrix semigroups and bounded languages. Information and Computation, 237:243-256, 2014. Google Scholar
  14. V. Chonev, J. Ouaknine, and J. Worrell. On the Complexity of the Orbit Problem. Journal of the ACM, 63(3):1-18, 2016. Google Scholar
  15. A. Condon and R. J. Lipton. On the complexity of space bounded interactive proofs. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science (FOCS), pages 462-467, 1989. Google Scholar
  16. L. Daviaud, M. Jurdzinski, R. Lazic, F. Mazowiecki, G. A. Pérez, and J. Worrell. When is containment decidable for probabilistic automata? In International Colloquium on Automata, Languages, and Programming (ICALP), pages 121:1-121:14, 2018. Google Scholar
  17. N. Fijalkow, C. Riveros, and J. Worrell. Probabilistic automata of bounded ambiguity. In 28th International Conference on Concurrency Theory (CONCUR), pages 19:1-19:14, 2017. Google Scholar
  18. V. Halava, J. Kari, and Y. Matiyasevich. On post correspondence problem for letter monotonic languages. Theoretical Computer Science, 410:30-32, 2009. Google Scholar
  19. M. Hirvensalo. Improved undecidability results on the emptiness problem of probabilistic and quantum cut-point languages. SOFSEM 2007: Theory and Practice of Computer Science, Lecture Notes in Computer Science, 4362:309-319, 2007. Google Scholar
  20. R. A. Horn and C. R. Johnson. Topics in matrix analysis. Cambridge University Press, 1991. Google Scholar
  21. O. Ibarra and B. Ravikumar. On sparseness, ambiguity and other decision problems for acceptors and transducers. In Proc. STACS 1986, volume 210, pages 171-179, 1986. Google Scholar
  22. R. Kannan and R. J. Lipton. Polynomial-time algorithm for the orbit problem. Journal of the ACM, 33(4):808-821, 1986. Google Scholar
  23. \BIBYu. Matiyasevich. Hilbert’s Tenth Problem. MIT Press, 1993. Google Scholar
  24. M. Mohri, F. Pereira, and M. Riley. Weighted finite-state transducers in speech recognition. Computer Speech &Language, 16(1):69-88, 2002. Google Scholar
  25. T. Neary. Undecidability in Binary Tag Systems and the Post Correspondence Problem for Five Pairs of Words. In STACS15, pages 649-661, 2015. Google Scholar
  26. A. Paz. Introduction to Probabilistic Automata. Academic Press, 1971. Google Scholar
  27. M. O. Rabin. Probabilistic automata. Information and Control, 6:230-245, 1963. Google Scholar
  28. C. Reutenauer. Propertiétés arithmétiques et topologiques de séries rationnelles en variables non commutatives. Thèse troisième cycle, Université Paris VI, 1977. Google Scholar
  29. A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, 1978. Google Scholar
  30. P. Turakainen. Generalized automata and stochastic languages. Proceedings of the American Mathematical Society, 21:303-309, 1969. Google Scholar
  31. A. Weber and H. Seidl. On the degree of ambiguity of finite automata. Theoretical Computer Science, 88(2):325-349, 1991. Google Scholar
  32. H. J. Woeginger and Z. Yu. On the equal-subset-sum problem. Information Processing Letters, 42(6):299-302, 1992. Google Scholar
  33. A. Yakaryilmaz and A. C. Say. Unbounded-error quantum computation with small space bounds. Information and Computation, 209(6):873-892, 2011. 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