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Semigroup Intersection Problems in the Heisenberg Groups

Author Ruiwen Dong

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LIPIcs.STACS.2023.25.pdf
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Author Details

Ruiwen Dong
  • Department of Computer Science, University of Oxford, UK

Funding

The author acknowledges support from UKRI Frontier Research Grant EP/X033813/1.

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Ruiwen Dong. Semigroup Intersection Problems in the Heisenberg Groups. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.25

Abstract

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups.
The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic manipulation
Keywords
  • semigroup intersection
  • orbit intersection
  • matrix semigroups
  • Heisenberg group
  • nilpotent groups

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References

  1. 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, pages 498-507, 1996. Google Scholar
  2. László Babai, Paolo Codenotti, Joshua A. Grochow, and Youming Qiao. Code equivalence and group isomorphism. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1395-1408. SIAM, 2011. Google Scholar
  3. Henry Frederick Baker. Alternants and continuous groups. Proceedings of the London Mathematical Society, 2(1):24-47, 1905. Google Scholar
  4. Gilbert Baumslag. Lecture notes on nilpotent groups. American Mathematical Society, 2007. Google Scholar
  5. 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, pages 187-206. SIAM, 2017. Google Scholar
  6. Paul C. Bell and Igor Potapov. On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups. International Journal of Foundations of Computer Science, 21(06):963-978, 2010. Google Scholar
  7. Paul C. Bell and Igor Potapov. On the computational complexity of matrix semigroup problems. Fundamenta Informaticae, 116(1-4):1-13, 2012. Google Scholar
  8. Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier. Decidable and undecidable problems about quantum automata. SIAM Journal on Computing, 34(6):1464-1473, 2005. Google Scholar
  9. John Edward Campbell. On a law of combination of operators (second paper). Proceedings of the London Mathematical Society, 1(1):14-32, 1897. Google Scholar
  10. Christian Choffrut and Juhani Karhumäki. Some decision problems on integer matrices. RAIRO-Theoretical Informatics and Applications-Informatique Théorique et Applications, 39(1):125-131, 2005. Google Scholar
  11. Henri Cohen. A course in computational algebraic number theory, volume 8. Springer-Verlag Berlin, 1993. Google Scholar
  12. Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On reachability problems for low-dimensional matrix semigroups. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 44:1-44:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.44.
  13. Harm Derksen, Emmanuel Jeandel, and Pascal Koiran. Quantum automata and algebraic groups. Journal of Symbolic Computation, 39(3-4):357-371, 2005. Google Scholar
  14. Ruiwen Dong. On the identity problem and the group problem for nilpotent groups, 2022. Submitted. URL: https://doi.org/10.48550/arXiv.2208.02164.
  15. Cornelia Druţu and Michael Kapovich. Geometric group theory, volume 63. American Mathematical Soc., 2018. Google Scholar
  16. Max Garzon and Yechezkel Zalcstein. On isomorphism testing of a class of 2-nilpotent groups. Journal of Computer and System Sciences, 42(2):237-248, 1991. URL: https://doi.org/10.1016/0022-0000(91)90012-T.
  17. Vesa Halava and Tero Harju. On markov’s undecidability theorem for integer matrices. In Semigroup Forum, volume 75, pages 173-180. Springer, 2007. Google Scholar
  18. Felix Hausdorff. Die symbolische Exponentialformel in der Gruppentheorie. Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 58:19-48, 1906. Google Scholar
  19. Derek F. Holt, Bettina Eick, and Eamonn A. O'Brien. Handbook of Computational Group Theory. Chapman and Hall/CRC, 2005. Google Scholar
  20. Roger Howe. On the role of the heisenberg group in harmonic analysis. Bulletin (New Series) of the American Mathematical Society, 3(2):821-843, 1980. Google Scholar
  21. 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, pages 530-539, 2018. Google Scholar
  22. Jun-ichi Igusa. Theta functions, volume 194. Springer Science & Business Media, 2012. Google Scholar
  23. Mikhail Ivanovich Kargapolov and Jurij Ivanovič Merzljakov. Fundamentals of the Theory of Groups, volume 62. Springer, 1979. Google Scholar
  24. Olga G. Kharlampovich and Mark V. Sapir. Algorithmic problems in varieties. International Journal of Algebra and Computation, 5(04n05):379-602, 1995. Google Scholar
  25. Aleksandr Aleksandrovich Kirillov. Lectures on the orbit method, volume 64. American Mathematical Soc., 2004. Google Scholar
  26. Sang-Ki Ko, Reino Niskanen, and Igor Potapov. On the identity problem for the special linear group and the heisenberg group. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 132:1-132:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.132.
  27. Daniel König, Markus Lohrey, and Georg Zetzsche. Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups. Algebra and Computer Science, 677:138-153, 2016. Google Scholar
  28. V. M. Kopytov. Solvability of the problem of occurrence in finitely generated soluble groups of matrices over the field of algebraic numbers. Algebra and Logic, 7(6):388-393, 1968. Google Scholar
  29. Hari Krovi and Martin Rötteler. An efficient quantum algorithm for the hidden subgroup problem over Weyl-Heisenberg groups. In Mathematical Methods in Computer Science, pages 70-88. Springer, 2008. Google Scholar
  30. Engel Lefaucheux. Private Communication, 2022. Google Scholar
  31. A. Markov. On certain insoluble problems concerning matrices. Doklady Akad. Nauk SSSR, 57(6):539-542, 1947. Google Scholar
  32. K. A. Mikhailova. The occurrence problem for direct products of groups. Matematicheskii Sbornik, 112(2):241-251, 1966. Google Scholar
  33. J. von Neumann. Die Eindeutigkeit der Schrödingerschen Operatoren. Mathematische Annalen, 104:570-578, 1931. URL: http://eudml.org/doc/159483.
  34. Michael S. Paterson. Unsolvability in 3 × 3 matrices. Studies in Applied Mathematics, 49(1):105-107, 1970. Google Scholar
  35. 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, pages 170-186. SIAM, 2017. Google Scholar
  36. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1998. Google Scholar
  37. Hermann Weyl. The theory of groups and quantum mechanics. Courier Corporation, 1950. Google Scholar
  38. Jae-Hyun Yang. Harmonic analysis on the quotient spaces of heisenberg groups. Nagoya mathematical journal, 123:103-117, 1991. Google Scholar
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