Skip to main content
SpringerLink
Log in
Book cover

International Conference on Current Trends in Theory and Practice of Informatics

SOFSEM 2017: SOFSEM 2017: Theory and Practice of Computer Science pp 268–279Cite as

Matrix Semigroup Freeness Problems in SL\((2,\mathbb {Z})\)

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10139))

Abstract

In this paper we study decidability and complexity of decision problems on matrices from the special linear group SL\((2,\mathbb {Z})\). In particular, we study the freeness problem: given a finite set of matrices G generating a multiplicative semigroup S, decide whether each element of S has at most one factorization over G. In other words, is G a code? We show that the problem of deciding whether a matrix semigroup in SL\((2,\mathbb {Z})\) is non-free is NP-hard. Then, we study questions about the number of factorizations of matrices in the matrix semigroup such as the finite freeness problem, the recurrent matrix problem, the unique factorizability problem, etc. Finally, we show that some factorization problems could be even harder in SL\((2,\mathbb {Z})\), for example we show that to decide whether every prime matrix has at most k factorizations is PSPACE-hard.

This research was supported by EPSRC grant EP/M00077X/1.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bell, P.C., Hirvensalo, M., Potapov, I.: Mortality for \(2\times 2\) matrices is NP-hard. In: Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, pp. 148–159 (2012)

    Google Scholar 

  2. Bell, P.C., Hirvensalo, M., Potapov, I.: The identity problem for matrix semigroups in SL\((2,\mathbb{Z})\) is NP-complete (2016). To appear in SODA 17

    Google Scholar 

  3. Bell, P.C., Potapov, I.: Periodic and infinite traces in matrix semigroups. In: Proceedings of the 34th Conference on Current Trends in Theory and Practice of Computer Science, pp. 148–161 (2008)

    Google Scholar 

  4. Bell, P.C., Potapov, I.: Reachability problems in quaternion matrix and rotation semigroups. Inf. Comput. 206(11), 1353–1361 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bell, P.C., Potapov, I.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bell, P.C., Potapov, I.: On the computational complexity of matrix semigroup problems. Fundamenta Infomaticae 116(1–4), 1–13 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Blondel, V.D., Cassaigne, J., Karhumäki, J.: Problem 10.3: freeness of multiplicative matrix semigroups. In: Unsolved Problems in Mathematical Systems and Control Theory, pp. 309–314. Princeton University Press (2004)

    Google Scholar 

  8. Cassaigne, J., Harju, T., Karhumäki, J.: On the undecidability of freeness of matrix semigroups. Int. J. Algebra Comput. 09(03–04), 295–305 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cassaigne, J., Nicolas, F.: On the decidability of semigroup freeness. RAIRO - Theor. Inf. Appl. 46(3), 355–399 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Charlier, E., Honkala, J.: The freeness problem over matrix semigroups and bounded languages. Inf. Comput. 237, 243–256 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Choffrut, C., Karhumäki, J.: Some decision problems on integer matrices. RAIRO - Theor. Inf. Appl. 39(1), 125–131 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elstrodt, J., Grunewald, F., Mennicke, J.: Arithmetic applications of the hyperbolic lattice point theorem. Proc. Lond. Math. Soc. s3–57(2), 239–283 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. García del Moral, M.P., Martín, I., Peña, J.M., Restuccia, A.: SL\((2, \mathbb{Z})\) symmetries, supermembranes and symplectic torus bundles. J. High Energy Phys. 9, 1–12 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Gawrychowski, P., Gutan, M., Kisielewicz, A.: On the problem of freeness of multiplicative matrix semigroups. Theor. Comput. Sci. 411(7–9), 1115–1120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gurevich, Y., Schupp, P.: Membership problem for the modular group. SIAM J. Comput. 37(2), 425–459 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Halava, V., Harju, T., Hirvensalo, M.: Undecidability bounds for integer matrices using claus instances. Int. J. Found. Comput. Sci. 18(05), 931–948 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Klarner, D.A., Birget, J.-C., Satterfield, W.: On the undecidability of the freeness of integer matrix semigroups. Int. J. Algebra Comput. 01(02), 223–226 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ko, S.-K., Potapov, I.: Matrix semigroup freeness problems in SL\((2,\mathbb{Z})\) (2016). CoRR, abs/1610.09834

    Google Scholar 

  19. Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)

    Google Scholar 

  20. Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977)

    MATH  Google Scholar 

  21. Mackenzie, D.: A New Twist in Knot Theory, vol. 7 (2009)

    Google Scholar 

  22. Mandel, A., Simon, I.: On finite semigroups of matrices. Theor. Comput. Sci. 5(2), 101–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. Noll, T.: Musical intervals and special linear transformations. J. Math. Music 1(2), 121–137 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Polterovich, L., Rudnick, Z.: Stable mixing for cat maps and quasi-morphisms of the modular group. Ergodic Theor. Dyn. Syst. 24, 609–619 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Potapov, I.: From post systems to the reachability problems for matrix semigroups and multicounter automata. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds.) DLT 2004. LNCS, vol. 3340, pp. 345–356. Springer, Heidelberg (2004). doi:10.1007/978-3-540-30550-7_29

    Chapter  Google Scholar 

  26. Potapov, I.: Composition problems for braids. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, vol. 24, pp. 175–187 (2013)

    Google Scholar 

  27. Potapov, I., Semukhin, P.: Vector reachability problem in SL\((2, \mathbb{Z})\). In: 41st International Symposium on Mathematical Foundations of Computer Science, pp. 84:1–84:14 (2016)

    Google Scholar 

  28. Rankin, R.: Modular Forms and Functions. Cambridge University Press, Cambridge (1977)

    Book  MATH  Google Scholar 

  29. Shallit, J.: A Second Course in Formal Languages and Automata Theory, 1st edn. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  30. Witten, E.: SL\((2, \mathbb{Z})\) action on three-dimensional conformal field theories with abelian symmetry, vol. 2, pp. 1173–1200 (2005)

    Google Scholar 

  31. Woeginger, G.J., Yu, Z.: On the equal-subset-sum problem. Inf. Process. Lett. 42(6), 299–302 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zagier, D.: Elliptic Modular Forms and Their Applications, pp. 1–103 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Ashton Street, Liverpool, L69 3BX, UK

    Sang-Ki Ko & Igor Potapov

Authors
  1. Sang-Ki Ko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sang-Ki Ko .

Editor information

Editors and Affiliations

  1. TU Dortmund , Dortmund, Germany

    Bernhard Steffen

  2. TU Dresden , Dresden, Germany

    Christel Baier

  3. Eindhoven University of Technology , Eindhoven, The Netherlands

    Mark van den Brand

  4. Alpen Adria University Klagenfurt , Klagenfurt, Austria

    Johann Eder

  5. Lero - Irish Software Research Center , Limerick, Ireland

    Mike Hinchey

  6. Lero - Irish Software Research Center , Limerick, Ireland

    Tiziana Margaria

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Ko, SK., Potapov, I. (2017). Matrix Semigroup Freeness Problems in SL\((2,\mathbb {Z})\) . In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-51963-0_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51962-3

  • Online ISBN: 978-3-319-51963-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation