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# On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond

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## Acknowledgements

We thank Prof. James Worrell for useful discussions, particularly related to S-unit equations.

## Cite As

Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.83

## Abstract

We consider the following variant of the Mortality Problem: given k x k matrices A_1, A_2, ...,A_{t}, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^{m_1} A_2^{m_2} * ... * A_{t}^{m_{t}} is equal to the zero matrix? It is known that this problem is decidable when t <= 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices. In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k <= 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m_1,m_2,m_3) for which the equation A_1^{m_1} A_2^{m_2} A_3^{m_3} equals the zero matrix is equal to a finite union of direct products of semilinear sets. For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 x 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computability
• Mathematics of computing → Computations on matrices
##### Keywords
• Linear recurrence sequences
• Skolem’s problem
• mortality problem
• matrix equations
• primary decomposition theorem
• Baker’s theorem

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## 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.
2. C. Baier, S. Kiefer, J. Klein, S. Klüppelholz, D. Müller, and J. Worrell. Markov Chains and Unambiguous Büchi Automata. In Computer Aided Verification - 28th International Conference, CAV 2016, Toronto, ON, Canada, July 17-23, 2016, Proceedings, Part I, pages 23-42, 2016.
3. P. C. Bell, V. Halava, T. Harju, J. Karhumäki, and I. Potapov. Matrix equations and Hilbert’s tenth problem. International Journal of Algebra and Computation, 18:1231-1241, 2008.
4. P. C. Bell, M. Hirvensalo, and I. Potapov. Mortality for 2× 2 matrices is NP-hard. In Mathematical Foundations of Computer Science (MFCS 2012), volume LNCS 7464, pages 148-159, 2012.
5. P. C. Bell, M. Hirvensalo, and I. Potapov. The identity problem for matrix semigroups in SL₂(ℤ) is NP-complete. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), pages 187-206, 2017.
6. Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond. CoRR, abs/1902.10188, 2019. URL: http://arxiv.org/abs/1902.10188.
7. V. Blondel, E. Jeandel, P. Koiran, and N. Portier. Decidable and undecidable problems about quantum automata. SIAM Journal on Computing, 34:6:1464-1473, 2005.
8. V. D. Blondel, O. Bournez, P. Koiran, C. Papadimitriou, and J. N. Tsitsiklis. Deciding stability and mortality of piecewise affine dynamical systems. Theoretical Computer Science, 255(1-2):687-696, 2001.
9. V. D. Blondel and N. Portier. The presence of a zero in an integer linear recurrent sequence is NP-hard to decide. Linear Algebra and its Applications, pages 91-98, 2002.
10. V. D. Blondel and J. N. Tsitsiklis. Complexity of stability and controllability of elementary hybrid systems. Automatica, 35:479-489, 1999.
11. Jin-yi Cai, Wolfgang H. J. Fuchs, Dexter Kozen, and Zicheng Liu. Efficient Average-Case Algorithms for the Modular Group. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20-22 November 1994, pages 143-152, 1994.
12. Jin-yi Cai, Richard J. Lipton, and Yechezkel Zalcstein. The Complexity of the Membership Problem for 2-generated Commutative Semigroups of Rational Matrices. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20-22 November 1994, pages 135-142, 1994.
13. J. Cassaigne, V. Halava, T. Harju, and F. Nicolas. Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More. CoRR, abs/1404.0644, 2014.
14. V. Chonev, J. Ouaknine, and J. Worrell. On the Complexity of the Orbit Problem. Journal of the ACM, 63(3):1-18, 2016.
15. Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the Skolem Problem for Continuous Linear Dynamical Systems. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 100:1-100:13, 2016.
16. J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman. S-unit equations and their applications. In New advances in transcendence theory (Durham, 1986), pages 110-174. Cambridge Univ. Press, Cambridge, 1988.
17. E. Galby, J. Ouaknine, and J. Worrell. On matrix powering in low dimensions. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS'15), pages 329-340, 2015.
18. K. Győry. On the abc conjecture in algebraic number fields. Acta Arith., 133(3):281-295, 2008.
19. V. Halava, T. Harju, M. Hirvensalo, and J. Karhumäki. Skolem’s problem - on the border between decidability and undecidability. In TUCS Technical Report Number 683, 2005.
20. G. Hansel. Une démonstration simple du théorème de Skolem-Mahler-Lech. Theoret. Comput. Sci., 43(1):91-98, 1986.
21. Michael A. Harrison. Lectures on Linear Sequential Machines. Academic Press, Inc., Orlando, FL, USA, 1969.
22. K. Hoffman and R. Kunze. Linear algebra. Second edition. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.
23. Juha Honkala. Products of matrices and recursively enumerable sets. Journal of Computer and System Sciences, 81(2):468-472, 2015.
24. R. Horn and C. Johnson. Matrix Analysis. Cambridge University Press, 1990.
25. Ravindran Kannan and Richard J. Lipton. The Orbit Problem is Decidable. In Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC '80, pages 252-261, New York, NY, USA, 1980. ACM.
26. J.-Y. Kao, N. Rampersad, and J. Shallit. On NFAs where all states are final, initial, or both. Theor. Comput. Sci., 410(47-49):5010-5021, 2009.
27. S.-K. Ko, R. Niskanen, and I. Potapov. On the Identity Problem for the Special Linear Group and the Heisenberg Group. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 132:1-132:15, 2018.
28. 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.
29. C. Lech. A note on recurring series. Ark. Mat. 2, 1953.
30. Alexei Lisitsa and Igor Potapov. Membership and Reachability Problems for Row-Monomial Transformations. In Jiří Fiala, Václav Koubek, and Jan Kratochvíl, editors, Mathematical Foundations of Computer Science 2004, pages 623-634, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.
31. Markus Lohrey. Rational subsets of unitriangular groups. IJAC, 25(1-2):113-122, 2015.
32. K. Mahler. Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen. In Akad. Wet. Amsterdam 38, pages 50-60, 1935.
33. M. Mignotte, T. N. Shorey, and R. Tijdeman. The distance between terms of an algebraic recurrence sequence. J. Reine Angew. Math., 349:63-76, 1984.
34. C. Nuccio and E. Rodaro. Mortality Problem for 2× 2 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: https://doi.org/10.1007/978-3-540-77566-9_34.
35. J. Ouaknine, J. Sousa Pinto, and J. Worrell. On termination of integer linear loops. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 957-969, 2015.
36. J. Ouaknine, A. Pouly, J. Sousa-Pinto, and J. Worrell. Solvability of Matrix-Exponential Equations. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS'16), pages 798-806, 2016.
37. Joël Ouaknine and James Worrell. Decision Problems for Linear Recurrence Sequences. In Reachability Problems - 6th International Workshop, RP 2012, Bordeaux, France, September 17-19, 2012. Proceedings, pages 21-28, 2012.
38. 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.
39. 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.
40. M. S. Paterson. Unsolvability in 3× 3 matrices. Studies in Applied Mathematics, 49(1):105-107, 1970.
41. 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 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 170-186, 2017.
42. 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.
43. T. Skolem. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen. Skand. Mat. Kongr., 8:163-188, 1934.
44. N. K. Vereshchagin. The problem of the appearance of a zero in a linear recursive sequence. Mat. Zametki 38, 347(2):609-615, 1985.