Document Open Access Logo

On Reachability Problems for Low-Dimensional Matrix Semigroups

Authors Thomas Colcombet , Joël Ouaknine , Pavel Semukhin , James Worrell

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2019.44.pdf
  • Filesize: 0.5 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Computing methodologies → Symbolic and algebraic algorithms
Keywords
  • membership problem
  • half-space reachability problem
  • matrix semigroups
  • Heisenberg group
  • general linear group

Metrics

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

Abstract

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

Cite As Get BibTex

Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On Reachability Problems for Low-Dimensional Matrix Semigroups. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.44

Author Details

Thomas Colcombet
  • IRIF, CNRS, Université Paris Diderot, France
Joël Ouaknine
  • The Max Planck Institute for Software Systems, Saarbrücken, Germany
  • Department of Computer Science, University of Oxford, United Kingdom
Pavel Semukhin
  • Department of Computer Science, University of Oxford, United Kingdom
James Worrell
  • Department of Computer Science, University of Oxford, United Kingdom

Funding

  • Colcombet, Thomas: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.670624), and by the DeLTA ANR project (ANR-16-CE40-0007).
  • Ouaknine, Joël: Supported by ERC grant AVS-ISS (648701) and by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Semukhin, Pavel: Supported by ERC grant AVS-ISS (648701).
  • Worrell, James: Supported by EPSRC Fellowship EP/N008197/1.

References

  1. László Babai. Trading Group Theory for Randomness. In Proceedings of the 17th Annual ACM Symposium on Theory of Computing, May 6-8, 1985, Providence, Rhode Island, USA, pages 421-429, 1985. Google Scholar
  2. László Babai, Robert Beals, Jin-yi Cai, Gábor Ivanyos, and Eugene M. Luks. Multiplicative Equations over Commuting Matrices. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '96, pages 498-507, Philadelphia, PA, USA, 1996. Society for Industrial and Applied Mathematics. Google Scholar
  3. Robert Beals. Algorithms for Matrix Groups and the Tits Alternative. J. Comput. Syst. Sci., 58(2):260-279, 1999. Google Scholar
  4. Paul Bell, Mika Hirvensalo, and Igor Potapov. The Identity Problem for Matrix Semigroups in SL(2,ℤ) is NP-complete. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, 2017. Google Scholar
  5. Paul C. Bell and Igor Potapov. On the Undecidability of the Identity Correspondence Problem and its Applications for Word and Matrix Semigroups. Int. J. Found. Comput. Sci., 21(6):963-978, 2010. Google Scholar
  6. Paul C. Bell and Igor Potapov. On the Computational Complexity of Matrix Semigroup Problems. Fundam. Inf., 116(1-4):1-13, 2012. Google Scholar
  7. V. Blondel, E. Jeandel, P. Koiran, and N. Portier. Decidable and Undecidable Problems about Quantum Automata. SIAM J. Comput., 34(6):1464-1473, 2005. URL: http://dx.doi.org/10.1137/S0097539703425861.
  8. Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. Google Scholar
  9. Julien Cassaigne, Vesa Halava, Tero Harju, and François Nicolas. Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More. CoRR, abs/1404.0644, 2014. URL: http://arxiv.org/abs/1404.0644.
  10. Christian Choffrut and Juhani Karhumäki. Some decision problems on integer matrices. RAIRO-Theor. Inf. Appl., 39(1):125-131, 2005. Google Scholar
  11. Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On reachability problems for low dimensional matrix semigroups. CoRR, abs/1902.09597, 2019. URL: http://arxiv.org/abs/1902.09597.
  12. H. Derksen, E. Jeandel, and P. Koiran. Quantum automata and algebraic groups. J. Symb. Comput., 39(3-4):357-371, 2005. Google Scholar
  13. Fritz J. Grunewald and Daniel Segal. How to solve a quadratic equation in integers. Mathematical Proceedings of the Cambridge Philosophical Society, 89(1):1–5, 1981. Google Scholar
  14. Fritz J. Grunewald and Daniel Segal. On the integer solutions of quadratic equations. J. Reine Angew. Math., 569:13-45, 2004. Google Scholar
  15. Yuri Gurevich and Paul Schupp. Membership Problem for the Modular Group. SIAM J. Comput., 37(2):425-459, May 2007. Google Scholar
  16. V. Halava and M. Hirvensalo. Improved matrix pair undecidability results. Acta Informatica, 44(3-4):191-205, 2007. Google Scholar
  17. Vesa Halava, Tero Harju, and Mika Hirvensalo. Undecidability Bounds for Integer Matrices Using Claus Instances. Int. J. Found. Comput. Sci., 18(5):931-948, 2007. Google Scholar
  18. B. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, volume 222 of Graduate Texts in Mathematics. Springer International Publishing, 2015. Google Scholar
  19. Ehud Hrushovski, Joël Ouaknine, Amaury Pouly, and James Worrell. Polynomial Invariants for Affine Programs. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 530-539, 2018. Google Scholar
  20. Ravindran Kannan and Richard J. Lipton. Polynomial-time algorithm for the orbit problem. J. ACM, 33(4):808-821, 1986. Google Scholar
  21. Felix Klaedtke and Harald Rueß. Parikh Automata and Monadic Second-Order Logics with Linear Cardinality Constraints. Technical report, Albert-Ludwigs-Universität Freiburg, 2002. Google Scholar
  22. Felix Klaedtke and Harald Rueß. Monadic Second-Order Logics with Cardinalities. In Automata, Languages and Programming, 30th International Colloquium, ICALP, pages 681-696, 2003. Google Scholar
  23. Sang-Ki Ko, Reino Niskanen, and Igor Potapov. On the Identity Problem for the Special Linear Group and the Heisenberg Group. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, pages 132:1-132:15, 2018. Google Scholar
  24. Daniel König, Markus Lohrey, and Georg Zetzsche. Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups. CoRR, abs/1507.05145, 2015. URL: http://arxiv.org/abs/1507.05145.
  25. Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. Google Scholar
  26. Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory. Dover Publications, Inc., New York, revised edition, 1976. Google Scholar
  27. A. Mandel and I. Simon. On Finite Semigroups of Matrices. Theor. Comput. Sci., 5(2):101-111, 1977. Google Scholar
  28. A. Markov. On certain insoluble problems concerning matrices. Doklady Akad. Nauk SSSR, 57(6):539-542, June 1947. Google Scholar
  29. K. A. Mihailova. The occurrence problem for a direct product of groups. Dokl. Akad. Nauk, 119:1103-1105, 1958. Google Scholar
  30. Joël Ouaknine and James Worrell. Positivity Problems for Low-Order Linear Recurrence Sequences. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 366-379, 2014. Google Scholar
  31. Michael S. Paterson. Unsolvability in 3× 3 matrices. Studies in Appl. Math., 49:105-107, 1970. Google Scholar
  32. Igor Potapov and Pavel Semukhin. Decidability of the Membership Problem for 2× 2 integer matrices. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 170-186, 2017. Google Scholar
  33. Igor Potapov and Pavel Semukhin. Membership Problem in GL(2, Z) Extended by Singular Matrices. In 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, pages 44:1-44:13, 2017. Google Scholar
  34. Igor Potapov and Pavel Semukhin. Vector and scalar reachability problems in SL(2,ℤ). J. Comput. Syst. Sci., 100:30-43, 2019. Google Scholar
  35. Robert A. Rankin. Modular forms and functions. Cambridge University Press, Cambridge-New York-Melbourne, 1977. Google Scholar
  36. Grzegorz Rozenberg and Arto Salomaa. Cornerstones of undecidability. Prentice Hall International Series in Computer Science. Prentice Hall, 1994. Google Scholar
  37. Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley &Sons, Inc., New York, NY, USA, 1986. Google Scholar
  38. Charles Sims. Computation with Finitely Presented Groups. Cambridge University Press, 1994. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact