Ackermannian Integer Compression and the Word Problem for Hydra Groups

Dison, Will; Einstein, Eduard; Riley, Tim

doi:10.4230/lipics.mfcs.2016.30

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2021

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“…Other examples where compression techniques lead to efficient algorithms in group theory can be found e.g. in [13,14] or [24,Theorems 4.6,4.8 and 4.9]. Finally, notice that in [28] the word search problem for the Baumslag group has been examined using parametrized complexity.…”

Section: Introductionmentioning

confidence: 99%

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

Mattes¹,

Weiss²

2021

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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 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.

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