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 \((x,y) \mapsto x\cdot 2^y\). 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\(\textemdash\)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.
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Acknowledgement
A conference version of this work has appeared at MFCS 2021 (Mattes & Weiß 2021). The second author (Armin Weiß) has been funded by DFG projects DI 435/7-1 and WE 6835/1-2.
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Mattes, C., Weiß, A. Parallel algorithms for power circuits and the word problem of the Baumslag group. comput. complex. 32, 10 (2023). https://doi.org/10.1007/s00037-023-00244-x
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DOI: https://doi.org/10.1007/s00037-023-00244-x