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

TL;DR

This paper explores parallel algorithms for power circuits and demonstrates that the word problem of the Baumslag group can be solved efficiently in NC, despite the complexity of integer comparison within power circuits.

Contribution

It shows that the Baumslag group's word problem, previously solvable in polynomial time, can also be solved in NC by leveraging the logarithmic depth of power circuits.

Findings

Word problem of Baumslag group is in NC.

Power circuits with logarithmic depth can be compared in NC.

Integer comparison in power circuits is P-complete in general.

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 - 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.