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Compression Techniques in Group Theory

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Connecting with Computability (CiE 2021)

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Abstract

This paper gives an informal overview over applications of compression techniques in algorithmic group theory.

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Notes

  1. 1.

    These are the one-relator groups \(\mathsf {BS}(p,q) = \langle a,t \mid t^{-1} a^p t = a^q \rangle \).

  2. 2.

    In fact, \(\mathsf {PSPACE}\)-hardness of the compressed word problem for \(G \wr \mathbb {Z}\) holds for a quite large class of non-solvable groups, namely all so-called uniformly SENS groups G [8], whereas for every non-abelian group G, the compressed word problem for \(G \wr \mathbb {Z}\) is already \(\mathsf {coNP}\)-hard [43].

  3. 3.

    \(\mathsf {coNP}\)-hardness holds for every uniformly SENS group G.

References

  1. Adjan, S.I.: The unsolvability of certain algorithmic problems in the theory of groups. Trudy Moskov. Mat. Obsc. 6, 231–298 (1957). in Russian

    MathSciNet  Google Scholar 

  2. Agol, I.: The virtual Haken conjecture. Documenta Mathematica 18, 1045–1087 (2013). With an appendix by Ian Agol, Daniel Groves, and Jason Manning

    Google Scholar 

  3. Agrawal, M., Biswas, S.: Primality and identity testing via Chinese remaindering. J. Assoc. Comput. Mach. 50(4), 429–443 (2003)

    Article  MathSciNet  Google Scholar 

  4. Artin, E.: Theorie der Zöpfe. Abh. Math. Semin. Univ. Hambg. 4(1), 47–72 (1925)

    Article  Google Scholar 

  5. Avenhaus, J., Madlener, K.: The Nielsen reduction and P-complete problems in free groups. Theoret. Comput. Sci. 32(1–2), 61–76 (1984)

    Article  MathSciNet  Google Scholar 

  6. Babai, L., Szemerédi, E.: On the complexity of matrix group problems I. In: Proceedings of the 25th Annual Symposium on Foundations of Computer Science, FOCS 1984, pp. 229–240 (1984)

    Google Scholar 

  7. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in \(\text{ NC}^1\). J. Comput. Syst. Sci. 38, 150–164 (1989)

    Article  MathSciNet  Google Scholar 

  8. Bartholdi, L., Figelius, M., Lohrey, M., Weiß, A.: Groups with ALOGTIME-hard word problems and PSPACE-complete circuit value problems. In: Proceedings of the 35th Computational Complexity Conference, CCC 2020. LIPIcs, vol. 169, pp. 29:1–29:29. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

    Google Scholar 

  9. Beaudry, M., McKenzie, P., Péladeau, P., Thérien, D.: Finite monoids: from word to circuit evaluation. SIAM J. Comput. 26(1), 138–152 (1997)

    Article  MathSciNet  Google Scholar 

  10. Birget, J.-C.: The groups of Richard Thompson and complexity. Int. J. Algebra Comput. 14(5–6), 569–626 (2004)

    Article  MathSciNet  Google Scholar 

  11. Boone, W.W.: The word problem. Ann. Math. Second Ser. 70, 207–265 (1959)

    Article  MathSciNet  Google Scholar 

  12. Cai, J.-Y.: Parallel computation over hyperbolic groups. In: Proceedings of the 24th Annual Symposium on Theory of Computing, STOC 1992, pp. 106–115. ACM Press (1992)

    Google Scholar 

  13. Cannonito, F.B.: Hierarchies of computable groups and the word problem. J. Symb. Log. 31, 376–392 (1966)

    Article  MathSciNet  Google Scholar 

  14. Charikar, M., et al.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)

    Article  MathSciNet  Google Scholar 

  15. Dehn, M.: Über unendliche diskontinuierliche Gruppen. Math. Ann. 71, 116–144 (1911). In German

    Article  MathSciNet  Google Scholar 

  16. Dehn, M.: Transformation der Kurven auf zweiseitigen Flächen. Math. Ann. 72, 413–421 (1912). In German

    Article  MathSciNet  Google Scholar 

  17. Diekert, V., Laun, J., Ushakov, A.: Efficient algorithms for highly compressed data: the word problem in Higman’s group is in P. Int. J. Algebra Comput. 22(8) (2012)

    Google Scholar 

  18. Diekert, V., Myasnikov, A., Weiß, A.: Conjugacy in Baumslag’s group, generic case complexity, and division in power circuits. Algorithmica 76(4), 961–988 (2016)

    Article  MathSciNet  Google Scholar 

  19. Dison, W., Einstein, E., Riley, T.R.: Taming the hydra: the word problem and extreme integer compression. Int. J. Algebra Comput. 28(7), 1299–1381 (2018)

    Article  MathSciNet  Google Scholar 

  20. Dison, W., Riley, T.R.: Hydra groups. Commentarii Mathematici Helvetici 88(3), 507–540 (2013)

    Article  MathSciNet  Google Scholar 

  21. Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett, Boston (1992)

    Book  Google Scholar 

  22. Figelius, M., Ganardi, M., Lohrey, M., Zetzsche, G.: The complexity of knapsack problems in wreath products. In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020. LIPIcs, vol. 168, pp. 126:1–126:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

    Google Scholar 

  23. Ge, G.: Testing equalities of multiplicative representations in polynomial time (extended abstract). In: Proceedings of the 34th Annual Symposium on Foundations of Computer Science, FOCS 1993, pp. 422–426 (1993)

    Google Scholar 

  24. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. MSRI, vol. 8, pp. 75–263. Springer, New York (1987). https://doi.org/10.1007/978-1-4613-9586-7_3

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

    Article  MathSciNet  Google Scholar 

  26. Haglund, F., Wise, D.T.: Coxeter groups are virtually special. Adv. Math. 224(5), 1890–1903 (2010)

    Article  MathSciNet  Google Scholar 

  27. Haubold, N., Lohrey, M.: Compressed word problems in HNN-extensions and amalgamated products. Theory Comput. Syst. 49(2), 283–305 (2011). https://doi.org/10.1007/s00224-010-9295-2

    Article  MathSciNet  MATH  Google Scholar 

  28. Haubold, N., Lohrey, M., Mathissen, C.: Compressed decision problems for graph products of groups and applications to (outer) automorphism groups. Int. J. Algebra Comput. 22(8) (2013)

    Google Scholar 

  29. Hertrampf, U., Lautemann, C., Schwentick, T., Vollmer, H., Wagner, K.W.: On the power of polynomial time bit-reductions. In: Proceedings of the 8th Annual Structure in Complexity Theory Conference, pp. 200–207. IEEE Computer Society Press (1993)

    Google Scholar 

  30. Hirshfeld, Y., Jerrum, M., Moller, F.: A polynomial algorithm for deciding bisimilarity of normed context-free processes. Theoret. Comput. Sci. 158(1 & 2), 143–159 (1996)

    Article  MathSciNet  Google Scholar 

  31. Holt, D.: Word-hyperbolic groups have real-time word problem. Int. J. Algebra Comput. 10, 221–228 (2000)

    Article  MathSciNet  Google Scholar 

  32. Holt, D., Lohrey, M., Schleimer, S.: Compressed decision problems in hyperbolic groups. In: Proceedings of the 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019. LIPIcs, vol. 126, pp. 37:1–37:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

    Google Scholar 

  33. Holt, D., Rees, S.: The compressed word problem in relatively hyperbolic groups. Technical report (2020). arxiv:2005.13917

  34. Ibarra, O.H., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. Assoc. Comput. Mach. 30(1), 217–228 (1983)

    Article  MathSciNet  Google Scholar 

  35. Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1–2), 1–46 (2004)

    Article  MathSciNet  Google Scholar 

  36. Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory, and random walks. J. Algebra 264(2), 665–694 (2003)

    Article  MathSciNet  Google Scholar 

  37. König, D., Lohrey, M.: Evaluation of circuits over nilpotent and polycyclic groups. Algorithmica 80(5), 1459–1492 (2018)

    Article  MathSciNet  Google Scholar 

  38. König, D., Lohrey, M.: Parallel identity testing for skew circuits with big powers and applications. Int. J. Algebra Comput. 28(6), 979–1004 (2018)

    Article  MathSciNet  Google Scholar 

  39. Lipton, R.J., Zalcstein, Y.: Word problems solvable in logspace. J. Assoc. Comput. Mach. 24(3), 522–526 (1977)

    Article  MathSciNet  Google Scholar 

  40. Lohrey, M.: Decidability and complexity in automatic monoids. Int. J. Found. Comput. Sci. 16(4), 707–722 (2005)

    Article  MathSciNet  Google Scholar 

  41. Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006)

    Article  MathSciNet  Google Scholar 

  42. Lohrey, M.: Algorithmics on SLP-compressed strings: a survey. Groups Complex. Cryptol. 4(2), 241–299 (2012)

    Article  MathSciNet  Google Scholar 

  43. Lohrey, M.: The Compressed Word Problem for Groups. SM, Springer, New York (2014). https://doi.org/10.1007/978-1-4939-0748-9

    Book  MATH  Google Scholar 

  44. Lohrey, M.: Subgroup membership in GL(2, Z). In: Proceedings of the 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021. LIPIcs, vol. 187, pp. 51:1–51:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  45. Lohrey, M., Schleimer, S.: Efficient computation in groups via compression. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 249–258. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74510-5_26

    Chapter  Google Scholar 

  46. Lohrey, M., Weiß, A.: The power word problem. In: Proceedings of the 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019. LIPIcs, vol. 138, pp. 43:1–43:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

    Google Scholar 

  47. Lohrey, M., Zetzsche, G.: Knapsack and the power word problem in solvable Baumslag-Solitar groups. In: Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020. LIPIcs, vol. 170, pp. 67:1–67:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

    Google Scholar 

  48. Macdonald, J.: Compressed words and automorphisms in fully residually free groups. Int. J. Algebra Comput. 20(3), 343–355 (2010)

    Article  MathSciNet  Google Scholar 

  49. Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106(1), 295–307 (1932)

    Article  MathSciNet  Google Scholar 

  50. Magnus, W.: On a theorem of Marshall Hall. Ann. Math. Second Ser. 40, 764–768 (1939)

    Article  MathSciNet  Google Scholar 

  51. Mattes, C., Weiß, A.: Parallel algorithms for power circuits and the word problem of the Baumslag group. CoRR, abs/2102.09921 (2021)

    Google Scholar 

  52. Mehlhorn, K., Sundar, R., Uhrig, C.: Maintaining dynamic sequences under equality-tests in polylogarithmic time. In: Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1994, pp. 213–222. ACM/SIAM (1994)

    Google Scholar 

  53. Miasnikov, A., Vassileva, S., Weiß, A.: The conjugacy problem in free solvable groups and wreath products of Abelian groups is in \(\text{ TC}^0\). Theory Comput. Syst. 63(4), 809–832 (2019)

    Article  MathSciNet  Google Scholar 

  54. Myasnikov, A., Ushakov, A., Won, D.W.: The word problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable. J. Algebra 345(1), 324–342 (2011)

    Article  MathSciNet  Google Scholar 

  55. Myasnikov, A.G., Ushakov, A., Won, D.W.: Power circuits, exponential algebra, and time complexity. Int. J. Algebra Comput. 22(6) (2012)

    Google Scholar 

  56. Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Am. Math. Soc. Transl. II Ser. 9, 1–122 (1958)

    Article  Google Scholar 

  57. Plandowski, W.: Testing equivalence of morphisms on context-free languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0049431

    Chapter  Google Scholar 

  58. Rabin, M.O.: Computable algebra, general theory and theory of computable fields. Trans. Am. Math. Soc. 95, 341–360 (1960)

    MathSciNet  MATH  Google Scholar 

  59. Robinson, D.: Parallel algorithms for group word problems. Ph.D. thesis, University of California, San Diego (1993)

    Google Scholar 

  60. Saxena, N.: Progress on polynomial identity testing-II. In: Agrawal, M., Arvind, V. (eds.) Perspectives in Computational Complexity. PCSAL, vol. 26, pp. 131–146. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05446-9_7

    Chapter  Google Scholar 

  61. Schleimer, S.: Polynomial-time word problems. Commentarii Mathematici Helvetici 83(4), 741–765 (2008)

    Article  MathSciNet  Google Scholar 

  62. Simon, H.-U.: Word problems for groups and contextfree recognition. In: Proceedings of Fundamentals of Computation Theory, FCT 1979, pp. 417–422. Akademie-Verlag (1979)

    Google Scholar 

  63. Waack, S.: The parallel complexity of some constructions in combinatorial group theory. J. Inf. Process. Cybern. EIK 26, 265–281 (1990)

    MathSciNet  MATH  Google Scholar 

  64. Wächter, J.P., Weiß, A.: An automaton group with PSPACE-complete word problem. In: Proceedings of the 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020. LIPIcs, vol. 154, pp. 6:1–6:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

    Google Scholar 

  65. Wehrfritz, B.A.F.: On finitely generated soluble linear groups. Mathematische Zeitschrift 170, 155–167 (1980)

    Article  MathSciNet  Google Scholar 

  66. Weiß, A.: A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups. In: Algebra and Computer Science, volume 677 of Contemporary Mathematics. American Mathematical Society (2016)

    Google Scholar 

  67. Wise, D.T.: The Structure of Groups with a Quasiconvex Hierarchy. Princeton University Press, Princeton (2021)

    Book  Google Scholar 

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Authors and Affiliations

  1. University of Siegen, Siegen, Germany

    Markus Lohrey

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  1. Markus Lohrey
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  1. University of Lille, Lille, France

    Liesbeth De Mol

  2. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

  3. University of Göttingen, Göttingen, Germany

    Florin Manea

  4. Ghent University, Ghent, Belgium

    David Fernández-Duque

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Lohrey, M. (2021). Compression Techniques in Group Theory. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_30

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