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Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won asa data structure for non-elementarily compressed integers supporting thearithmetic operations addition and $(x,y) \mapsto x\cdot 2^y$. The same authorsapplied power circuits to give a polynomial-time solution to the word problemof the Baumslag group, which has a non-elementary Dehn function.  In this work, we examine power circuits and the word problem of the Baumslaggroup under parallel complexity aspects. In particular, we establish that theword problem of the Baumslag group can be solved in NC - even though one of theessential steps is to compare two integers given by power circuits and this, ingeneral, is shown to be P-complete. The key observation is that the depth ofthe occurring power circuits is logarithmic and such power circuits can becompared in NC.

Parallel algorithms for power circuits and the word problem of the Baumslag group