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 $(x,y) \mapsto x2^y$. 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 essentials steps is to compare two integers given by power circuits and this problem, in general, is P-complete. The key observation here is that the depth of the occurring power circuits is logarithmic and such power circuits, indeed, can be compared in NC.

中文翻译：

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电源电路的并行算法和Baumslag组的单词问题

电源电路由Myasnikov，Ushakov和Won于2012年引入，作为非基本压缩整数的数据结构，支持算术加法和$（x，y）\ mapsto x2 ^ y $。同一作者应用电源电路为具有非基本Dehn函数的Baumslag组的单词问题提供了多项式时间解。在这项工作中，我们研究了并行复杂度方面的电源电路和Baumslag组的词问题。特别是，我们确定可以在NC中解决Baumslag组的单词问题-即使基本步骤之一是比较电源电路给出的两个整数，并且这个问题通常是P完全的。此处的主要观察结果是，发生的电源电路的深度是对数的，而实际上，