Membership Problem in GL(2, Z) Extended by Singular Matrices
We consider the membership problem for matrix semigroups, which is the problem to decide whether a matrix belongs to a given finitely generated matrix semigroup.
In general, the decidability and complexity of this problem for two-dimensional matrix semigroups remains open. Recently there was a significant progress with this open problem by showing that the membership is decidable for 2x2 nonsingular integer matrices. In this paper we focus on the membership for singular integer matrices and prove that this problem is decidable for 2x2 integer matrices whose determinants are equal to 0, 1, -1 (i.e. for matrices from GL(2,Z) and any singular matrices). Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words and conversion of the membership problem into decision problem on regular languages.
Matrix Semigroups
Membership Problem
General Linear Group
Singular Matrices
Automata and Formal Languages
44:1-44:13
Regular Paper
Igor
Potapov
Igor Potapov
Pavel
Semukhin
Pavel Semukhin
10.4230/LIPIcs.MFCS.2017.44
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.
Paul Bell and Igor Potapov. On undecidability bounds for matrix decision problems. Theoretical Computer Science, 391(1-2):3-13, 2008.
Paul C. Bell, Mika Hirvensalo, and Igor Potapov. Mortality for 2x2 matrices is NP-hard. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012, volume 7464 of Lecture Notes in Computer Science, pages 148-159. Springer Berlin Heidelberg, 2012.
Paul C. Bell, Mika Hirvensalo, and Igor Potapov. The identity problem for matrix semigroups in SL₂(ℤ) is NP-complete. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 187-206, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.13.
http://dx.doi.org/10.1137/1.9781611974782.13
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.
Paul C. Bell and Igor Potapov. On the computational complexity of matrix semigroup problems. Fundam. Inf., 116(1-4):1-13, January 2012.
Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier. Decidable and undecidable problems about quantum automata. SIAM J. Comput., 34(6):1464-1473, June 2005.
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.
Christian Choffrut and Juhani Karhumaki. Some decision problems on integer matrices. RAIRO-Theor. Inf. Appl., 39(1):125-131, 2005.
J. Esparza, A. Finkel, and R. Mayr. On the verification of broadcast protocols. In Logic in Computer Science, 1999. Proceedings. 14th Symposium on, pages 352-359, 1999.
Esther Galby, Joël Ouaknine, and James Worrell. On Matrix Powering in Low Dimensions. 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 329-340, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Yuri Gurevich and Paul Schupp. Membership problem for the modular group. SIAM J. Comput., 37(2):425-459, May 2007.
Vesa Halava, Tero Harju, Mika Hirvensalo, and Juhani Karhumaki. Skolem’s problem - on the border between decidability and undecidability. Technical Report 683, Turku Centre for Computer Science, 2005.
R. Kannan and R. J. Lipton. Polynomial-time algorithm for the orbit problem. J. ACM, 33(4):808-821, August 1986. URL: http://dx.doi.org/10.1145/6490.6496.
http://dx.doi.org/10.1145/6490.6496
Ravindran Kannan and Achim Bachem. Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput., 8(4):499-507, 1979.
Alexei Lisitsa and Igor Potapov. Membership and reachability problems for row-monomial transformations. In Mathematical Foundations of Computer Science 2004, 29th International Symposium, MFCS 2004, Prague, Czech Republic, August 22-27, 2004, Proceedings, pages 623-634, 2004.
Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89.
Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory. Dover Publications, Inc., New York, revised edition, 1976. Presentations of groups in terms of generators and relations.
C. Nuccio and Emanuele Rodaro. Mortality problem for 2times2 integer matrices. In SOFSEM 2008: Theory and Practice of Computer Science, 34th Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 19-25, 2008, Proceedings, pages 400-405, 2008. URL: http://dx.doi.org/10.1007/978-3-540-77566-9_34.
http://dx.doi.org/10.1007/978-3-540-77566-9_34
Joël Ouaknine, João Sousa Pinto, and James Worrell. On termination of integer linear loops. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 957-969. SIAM, 2015.
Joël Ouaknine and James Worrell. On the positivity problem for simple linear recurrence sequences,. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 318-329, 2014.
Joël Ouaknine and James Worrell. Ultimate positivity is decidable for simple linear recurrence sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 330-341, 2014.
M. S. Paterson. Unsolvability in 3times3 matrices. Studies in Applied Mathematics, 49(1):pp.105-107, 1970.
Igor Potapov and Pavel Semukhin. Vector reachability problem in SL(2,Z). In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 84:1-84:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.84.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.84
Igor Potapov and Pavel Semukhin. Decidability of the membership problem for 2times2 integer matrices. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 170-186, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.12.
http://dx.doi.org/10.1137/1.9781611974782.12
Robert A. Rankin. Modular forms and functions. Cambridge University Press, Cambridge-New York-Melbourne, 1977.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode