A logspace solution to the word and conjugacy
                    problem of generalized Baumslag-Solitar
                    groups

Weiß, Armin

doi:10.1090/conm/677/13628

Cited by 13 publications

(10 citation statements)

References 51 publications

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“…The first result in this direction was proved by Lipton and Zalcstein [24] (if the field F has characteristic zero) and Simon [38] (if the field F has prime characteristic): if G is a finitely generated linear group over an arbitrary field F (i.e., a finitely generated group of invertible matrices over F ), then the word problem for G can be solved in deterministic logarithmic space. Related results can be found in [20,43].…”

Section: Introductionmentioning

confidence: 67%

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Streaming word problems

Lohrey¹,

Lück²

2022

Preprint

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We study deterministic and randomized streaming algorithms for word problems of finitely generated groups. For finitely generated linear groups, metabelian groups and free solvable groups we show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems. We also show that the class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group theoretical constructions: finite extensions, direct products, free products and wreath products by free abelian groups. We contrast these results with several lower bound. An example of a finitely presented group, where the word problem has only a linear space randomized streaming algorithm, is Thompson's group F .

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Section: Introductionmentioning

confidence: 67%

“…An example of group that is not residually finite is the Baumslag-Solitar group BS(2, 3). The word problem for every Baumslag-Solitar group BS(p, q) can be solved in logarithmic space [43].…”

Section: Open Problemsmentioning

confidence: 99%

Streaming word problems

Lohrey¹,

Lück²

2022

Preprint

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“…x ≡ k ℓ mod K ℓ (here congruent modulo ∞ means equality). Since the K ℓ are all β-smooth, they can be factored in TC 0 and a solution (if there is one) of this system can be determined in TC 0 (see e. g. [29,Lem. 27]) with the help of Hesse's division circuit [7,8] using the Chinese remainder theorem.…”

Section: Otherwise (And Likewise For G)mentioning

confidence: 99%

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

2018

Self Cite

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We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC 0 -Turingreducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC 0 -many-one-reducible to the conjugacy problem in A ≀ B.Furthermore, under certain natural conditions, we give a uniform TC 0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC 0 solution to the conjugacy problem in iterated wreath products of abelian groups -and, by the Magnus embedding, also in free solvable groups.

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“…Note that the conjugacy decision problem over BS (1,2) resides in the complexity class TC 0 [27], where TC 0 is the class of constant-depth arithmetic circuits using AND, OR, NOT, and majority gates.…”

Section: Test Groupsmentioning

confidence: 99%

Solving the Conjugacy Decision Problem via Machine Learning

Gryak

Haralick

Kahrobaei

2018

Experimental Mathematics

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Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography.As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and Ntuple neural network models are evaluated for several non-free groups. The very high accuracy of these classifiers suggests an underlying mathematical relationship with respect to conjugacy in the tested groups.

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A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups

Cited by 13 publications

References 51 publications

Streaming word problems

Streaming word problems

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

Solving the Conjugacy Decision Problem via Machine Learning

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