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Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis

Author Armin Weiß

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • equations in groups
  • solvable groups
  • exponential time hypothesis

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Abstract

Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in 𝖯 for nilpotent groups while it is 𝖭𝖯-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in 𝖯 for any group of Fitting length at least four and for certain groups of Fitting length three. Moreover, the same hardness result applies to the equation identity problem.

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Armin Weiß. Hardness of Equations over Finite Solvable Groups Under the Exponential Time Hypothesis. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 102:1-102:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.102

Author Details

Armin Weiß
  • Universität Stuttgart, Institut für Formale Methoden der Informatik (FMI), Germany

Funding

  • Weiß, Armin: Funded by DFG project DI 435/7-1.

Acknowledgements

I am grateful to Moses Ganardi for bringing my attention both to the AND-weakness conjecture and to the exponential time hypothesis. I am also thankful to David A. Mix Barrington for an interesting email exchange concerning the AND-weakness conjecture and the idea to include steps of the lower central series in Proposition 8 to get a more refined upper bound. Furthermore, I am indebted to Caroline Mattes and Jan Philipp Wächter for many helpful discussions. Finally, I want to thank the anonymous referees for their valuable comments.

References

  1. Scott Aaronson, Russell Impagliazzo, and Dana Moshkovitz. AM with multiple merlins. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 44-55. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.13.
  2. Jorge Almeida, M. V. Volkov, and S. V. Goldberg. Complexity of the identity checking problem for finite semigroups. Journal of Mathematical Sciences, 158(5):605-614, 2009. URL: https://doi.org/10.1007/s10958-009-9397-z.
  3. David A. Mix Barrington. Width-3 permutation branching programs. Technical Report TM-293, MIT Laboratory for Computer Science, 1985. Google Scholar
  4. David A. Mix Barrington, Richard Beigel, and Steven Rudich. Representing Boolean functions as polynomials modulo composite numbers. Computational Complexity, 4:367-382, 1994. URL: https://doi.org/10.1007/BF01263424.
  5. David A. Mix Barrington, Pierre McKenzie, Cristopher Moore, Pascal Tesson, and Denis Thérien. Equation satisfiability and program satisfiability for finite monoids. In Mathematical Foundations of Computer Science 2000, 25th International Symposium, MFCS 2000, Proceedings, volume 1893 of Lecture Notes in Computer Science, pages 172-181. Springer, 2000. URL: https://doi.org/10.1007/3-540-44612-5_13.
  6. David A. Mix Barrington, Howard Straubing, and Denis Thérien. Non-uniform automata over groups. Inf. Comput., 89(2):109-132, 1990. URL: https://doi.org/10.1016/0890-5401(90)90007-5.
  7. Mark Braverman, Young Kun-Ko, Aviad Rubinstein, and Omri Weinstein. ETH hardness for densest-k-subgraph with perfect completeness. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1326-1341. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.86.
  8. Mark Braverman, Young Kun-Ko, and Omri Weinstein. Approximating the best nash equilibrium in n^o^(log n)-time breaks the exponential time hypothesis. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 970-982. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.66.
  9. Stanley Burris and J. Lawrence. Results on the equivalence problem for finite groups. Algebra Universalis, 52(4):495-500 (2005), 2004. URL: https://doi.org/10.1007/s00012-004-1895-8.
  10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  11. Volker Diekert and Murray Elder. Solutions of twisted word equations, EDT0L languages, and context-free groups. In ICALP 2017, Proceedings, volume 80 of LIPIcs, pages 96:1-96:14, Dagstuhl, Germany, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.96.
  12. Attila Földvári. The complexity of the equation solvability problem over semipattern groups. IJAC, 27(2):259, 2017. URL: https://doi.org/10.1142/S0218196717500126.
  13. Attila Földvári and Gábor Horváth. The complexity of the equation solvability and equivalence problems over finite groups. International Journal of Algebra and Computation, 30(03):607-623, 2020. URL: https://doi.org/10.1142/S0218196720500137.
  14. Albert Garreta, Alexei Miasnikov, and Denis Ovchinnikov. Diophantine problems in solvable groups. Bulletin of Mathematical Sciences, January 2020. URL: https://doi.org/10.1142/S1664360720500058.
  15. Mikael Goldmann and Alexander Russell. The complexity of solving equations over finite groups. Inf. Comput., 178(1):253-262, 2002. URL: https://doi.org/10.1006/inco.2002.3173.
  16. Tomasz A. Gorazd and Jacek Krzaczkowski. Term equation satisfiability over finite algebras. IJAC, 20(8):1001-1020, 2010. URL: https://doi.org/10.1142/S021819671000600X.
  17. Kristoffer Arnsfelt Hansen and Michal Koucký. A new characterization of ACC⁰ and probabilistic CC⁰. Computational Complexity, 19(2):211-234, 2010. URL: https://doi.org/10.1007/s00037-010-0287-z.
  18. Gábor Horváth. The complexity of the equivalence and equation solvability problems over nilpotent rings and groups. Algebra Universalis, 66(4):391-403, 2011. URL: https://doi.org/10.1007/s00012-011-0163-y.
  19. Gábor Horváth. The complexity of the equivalence and equation solvability problems over meta-Abelian groups. J. Algebra, 433:208-230, 2015. URL: https://doi.org/10.1016/j.jalgebra.2015.03.015.
  20. Gábor Horváth and Csaba Szabó. The extended equivalence and equation solvability problems for groups. Discrete Math. Theor. Comput. Sci., 13(4):23-32, 2011. Google Scholar
  21. Gábor Horváth and Csaba Szabó. Equivalence and equation solvability problems for the alternating group 𝐀₄. J. Pure Appl. Algebra, 216(10):2170-2176, 2012. URL: https://doi.org/10.1016/j.jpaa.2012.02.007.
  22. Gábor Horváth and Csaba A. Szabó. The complexity of checking identities over finite groups. IJAC, 16(5):931-940, 2006. URL: https://doi.org/10.1142/S0218196706003256.
  23. B. Huppert. Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin-New York, 1967. Google Scholar
  24. Pawel M. Idziak, Piotr Kawalek, and Jacek Krzaczkowski. Intermediate problems in modular circuits satisfiability. In LICS 2020, Proceedings. ACM, 2020. Preprint at https://arxiv.org/abs/2002.08626. URL: https://doi.org/10.1145/3373718.3394780.
  25. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  26. Marcel Jackson and Ralph McKenzie. Interpreting graph colorability in finite semigroups. IJAC, 16(1):119-140, 2006. URL: https://doi.org/10.1142/S0218196706002846.
  27. Andrzej Kisielewicz. Complexity of semigroup identity checking. IJAC, 14(4):455-464, 2004. URL: https://doi.org/10.1142/S0218196704001840.
  28. Ondrej Klíma, Pascal Tesson, and Denis Thérien. Dichotomies in the complexity of solving systems of equations over finite semigroups. Theory Comput. Syst., 40(3):263-297, 2007. URL: https://doi.org/10.1007/s00224-005-1279-2.
  29. Ondřej Klíma. Complexity issues of checking identities in finite monoids. Semigroup Forum, 79(3):435-444, 2009. URL: https://doi.org/10.1007/s00233-009-9180-y.
  30. Michael Kompatscher. CC-circuits and the expressive power of nilpotent algebras. CoRR, abs/1911.01479, 2019. URL: http://arxiv.org/abs/1911.01479.
  31. Michael Kompatscher. Notes on extended equation solvability and identity checking for groups. Acta Math. Hungar., 159(1):246-256, 2019. URL: https://doi.org/10.1007/s10474-019-00924-7.
  32. Markus Lohrey and Géraud Sénizergues. Theories of HNN-extensions and amalgamated products. In ICALP 2006, Proceedings, pages 504-515, 2006. URL: https://doi.org/10.1007/11787006_43.
  33. Gennadií Semyonovich Makanin. The problem of solvability of equations in a free semigroup. Math. Sbornik, 103:147-236, 1977. English transl. in Math. USSR Sbornik 32 (1977). Google Scholar
  34. Pierre McKenzie, Pierre Péladeau, and Denis Thérien. NC¹: The automata-theoretic viewpoint. Computational Complexity, 1:330-359, 1991. URL: https://doi.org/10.1007/BF01212963.
  35. John L. Rhodes and Benjamin Steinberg. The 𝔮-theory of finite semigroups. Springer Monographs in Mathematics. Springer, 2009. Google Scholar
  36. Derek J. S. Robinson. A course in the theory of groups, volume 80 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1996. URL: https://doi.org/10.1007/978-1-4419-8594-1.
  37. Vitaly Roman'kov. Equations in free metabelian groups. Siberian Mathematical Journal, 20, May 1979. URL: https://doi.org/10.1007/BF00969959.
  38. Steve Seif. The Perkins semigroup has co-NP-complete term-equivalence problem. Internat. J. Algebra Comput., 15(2):317-326, 2005. URL: https://doi.org/10.1142/S0218196705002293.
  39. Steve Seif and Csaba Szabó. Computational complexity of checking identities in 0-simple semigroups and matrix semigroups over finite fields. Semigroup Forum, 72(2):207-222, 2006. URL: https://doi.org/10.1007/s00233-005-0510-4.
  40. Csaba Szabó and Vera Vértesi. The complexity of checking identities for finite matrix rings. Algebra Universalis, 51(4):439-445, 2004. URL: https://doi.org/10.1007/s00012-004-1873-1.
  41. Yanior Weg. Normal subgroup of Fitting length i contained in i-th term of upper Fitting series? (answer). MathOverflow. URL: https://mathoverflow.net/questions/350552/ (visited on: 2020-04-24). URL: https://mathoverflow.net/questions/350552/.
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