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# Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group

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## Cite As

Caroline Mattes and Armin Weiß. Parallel Algorithms for Power Circuits and the Word Problem of the Baumslag Group. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 74:1-74:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.74

## Abstract

Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x,y) ↦ x⋅2^y. The same authors applied power circuits to give a polynomial-time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function. In this work, we examine power circuits and the word problem of the Baumslag group under parallel complexity aspects. In particular, we establish that the word problem of the Baumslag group can be solved in NC - even though one of the essential steps is to compare two integers given by power circuits and this, in general, is shown to be 𝖯-complete. The key observation is that the depth of the occurring power circuits is logarithmic and such power circuits can be compared in NC.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
• Theory of computation → Circuit complexity
##### Keywords
• Word problem
• Baumslag group
• power circuit
• parallel complexity

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## References

1. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009.
2. Owen Baker. The conjugacy problem for Higman’s group. Internat. J. Algebra Comput., 30(6):1211-1235, 2020. URL: https://doi.org/10.1142/S0218196720500393.
3. G. Baumslag, A. G. Myasnikov, and V. Shpilrain. Open problems in combinatorial group theory. Second Edition. In Combinatorial and geometric group theory, volume 296 of Contemporary Mathematics, pages 1-38. American Mathematical Society, 2002.
4. Gilbert Baumslag. A non-cyclic one-relator group all of whose finite quotients are cyclic. Journal of the Australian Mathematical Society, 10(3-4):497-498, 1969.
5. W. W. Boone. The Word Problem. Ann. of Math., 70(2):207-265, 1959.
6. John L. Britton. The word problem. Ann. of Math., 77:16-32, 1963.
7. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. The MIT Press, 3 edition, 2009.
8. Max Dehn. Ueber unendliche diskontinuierliche Gruppen. Math. Ann., 71:116-144, 1911.
9. Volker Diekert, Jürn Laun, and Alexander Ushakov. Efficient algorithms for highly compressed data: The word problem in Higman’s group is in P. In Proc. 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, Paris, France, volume 14 of LIPIcs, pages 218-229. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.218.
10. Volker Diekert, Jürn Laun, and Alexander Ushakov. Efficient algorithms for highly compressed data: The word problem in Higman’s group is in P. International Journal of Algebra and Computation, 22(8), 2013. URL: https://doi.org/10.1142/S0218196712400085.
11. Volker Diekert, Alexei G. Myasnikov, and Armin Weiß. Conjugacy in Baumslag’s Group, Generic Case Complexity, and Division in Power Circuits. In Alberto Pardo and Alfredo Viola, editors, Latin American Theoretical Informatics Symposium, volume 8392 of Lecture Notes in Computer Science, pages 1-12. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_1.
12. Volker Diekert, Alexei G. Myasnikov, and Armin Weiß. Conjugacy in Baumslag’s group, generic case complexity, and division in power circuits. Algorithmica, 74:961-988, 2016. URL: https://doi.org/10.1007/s00453-016-0117-z.
13. Will Dison, Eduard Einstein, and Timothy R. Riley. Ackermannian integer compression and the word problem for hydra groups. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 30:1-30:14, 2016. URL: https://doi.org/10.4230/LIPIcs.MFCS.2016.30.
14. Will Dison, Eduard Einstein, and Timothy R. Riley. Taming the hydra: The word problem and extreme integer compression. Int. J. Algebra Comput., 28(7):1299-1381, 2018. URL: https://doi.org/10.1142/S0218196718500583.
15. S. M. Gersten. Isodiametric and isoperimetric inequalities in group extensions. Preprint, 1991.
16. Graham Higman. A finitely generated infinite simple group. J. London Math. Soc., 26:61-64, 1951.
17. I. Kapovich, A. G. Miasnikov, P. Schupp, and V. Shpilrain. Generic-case complexity, decision problems in group theory and random walks. Journal of Algebra, 264:665-694, 2003.
18. Jonathan Kausch. The parallel complexity of certain algorithmic problems in group theory. Dissertation, Institut für Formale Methoden der Informatik, Universität Stuttgart, 2017.
19. Daniel König and Markus Lohrey. Evaluation of circuits over nilpotent and polycyclic groups. Algorithmica, 80(5):1459-1492, 2018. URL: https://doi.org/10.1007/s00453-017-0343-z.
20. Jürn Laun. Efficient algorithms for highly compressed data: The word problem in generalized Higman groups is in P. Theory Comput. Syst., 55(4):742-770, 2014. URL: https://doi.org/10.1007/s00224-013-9509-5.
21. J. Lehnert and P. Schweitzer. The co-word problem for the Higman-Thompson group is context-free. Bull. London Math. Soc., 39:235-241, 2007. URL: https://doi.org/10.1112/blms/bdl043.
22. Richard J. Lipton and Yechezkel Zalcstein. Word problems solvable in logspace. J. ACM, 24:522-526, 1977.
23. Markus Lohrey. Decidability and complexity in automatic monoids. International Journal of Foundations of Computer Science, 16(4):707-722, 2005.
24. Markus Lohrey. The Compressed Word Problem for Groups. Springer Briefs in Mathematics. Springer, 2014. URL: https://doi.org/10.1007/978-1-4939-0748-9.
25. Roger Lyndon and Paul Schupp. Combinatorial Group Theory. Classics in Mathematics. Springer, 2001. First edition 1977.
26. Wilhelm Magnus. Das Identitätsproblem für Gruppen mit einer definierenden Relation. Mathematische Annalen, 106:295-307, 1932.
27. Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial Group Theory. Dover, 2004.
28. Caroline Mattes and Armin Weiß. Parallel algorithms for power circuits and the word problem of the Baumslag group. CoRR, abs/2102.09921, 2021. URL: http://arxiv.org/abs/2102.09921.
29. Alexei Miasnikov and Andrey Nikolaev. On parameterized complexity of the word search problem in the Baumslag-Gersten group. In ISSAC '20: International Symposium on Symbolic and Algebraic Computation, Kalamata, Greece, July 20-23, 2020, pages 360-363, 2020. URL: https://doi.org/10.1145/3373207.3404042.
30. Alexei G. Myasnikov, Alexander Ushakov, and Won Dong-Wook. The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable. Journal of Algebra, 345:324-342, 2011. URL: http://www.sciencedirect.com/science/article/pii/S0021869311004492.
31. Alexei G. Myasnikov, Alexander Ushakov, and Won Dong-Wook. Power circuits, exponential algebra, and time complexity. International Journal of Algebra and Computation, 22(6):3-53, 2012.
32. Alexei G. Myasnikov and Sasha Ushakov. Cryptography and groups (CRAG). Software Library. URL: http://www.stevens.edu/algebraic/downloads.php.
33. P. S. Novikov. On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov, pages 1-143, 1955. In Russian.
34. A. N. Platonov. Isoparametric function of the Baumslag-Gersten group. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3:12-17, 2004. Russian. Engl. transl. Moscow Univ. Math. Bull. 59 (3) (2004), 12-17.
35. David Robinson. Parallel Algorithms for Group Word Problems. PhD thesis, University of California, San Diego, 1993.
36. Mark V. Sapir, Jean-Camille Birget, and Eliyahu Rips. Isoperimetric and Isodiametric Functions of Groups. Ann. Math., 156(2):345-466, 2002.
37. A. L. Semenov. Logical theories of one-place functions on the natural number series. Izv. Akad. Nauk SSSR Ser. Mat., 47(3):623-658, 1983.
38. Hans-Ulrich Simon. Word problems for groups and contextfree recognition. In Proceedings of Fundamentals of Computation Theory (FCT'79), Berlin/Wendisch-Rietz (GDR), pages 417-422. Akademie-Verlag, 1979.
39. Michael Sipser. Introduction to the Theory of Computation. International Thomson Publishing, 1st edition, 1996.
40. Heribert Vollmer. Introduction to Circuit Complexity. Springer, Berlin, 1999.
41. Armin Weiß. On the Complexity of Conjugacy in Amalgamated Products and HNN Extensions. Dissertation, Institut für Formale Methoden der Informatik, Universität Stuttgart, 2015.
42. Armin Weiß. A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups. In Algebra and Computer Science, volume 677 of Contemporary Mathematics, pages 185-212. American Mathematical Society, 2016.
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