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Compressed word problems in HNN-extensions and amalgamated products [article]

Niko Haubold, Markus Lohrey
2008 arXiv   pre-print
It is shown that the compressed word problem for an HNN-extension with base group H and finite associated subgroups is polynomial time Turing-reducible to the compressed word problem for H.  ...  An analogous result for amalgamated free products is shown as well.  ...  In this paper, we prove a transfer theorem for the compressed word problem of HNN-extensions [7] .  ... 
arXiv:0811.3303v1 fatcat:ph4em5tysnfm3mqqqjm253gtjy

Compressed Word Problems in HNN-extensions and Amalgamated Products

Niko Haubold, Markus Lohrey
2010 Theory of Computing Systems  
It is not hard to see that the word problem for an HNN-extension (1) with A finite can be reduced in polynomial time to the word problem of the base group H.  ...  In this paper, we restrict to HNN-extensions (resp. amalgamated products) with finite associated (resp. amalgamated) subgroups, which is an important subcase.  ...  From the close relationship of HNN-extensions with amalgamated free products, a polynomial time reduction from the compressed problem for an amalgamated free product H 1 * H 2 | a = ϕ(a) (a ∈ A 1 ) (with  ... 
doi:10.1007/s00224-010-9295-2 fatcat:bvn4l2vyvfdnblg3bdsedthnsy

Compressed Word Problems in HNN-Extensions and Amalgamated Products [chapter]

Niko Haubold, Markus Lohrey
2009 Lecture Notes in Computer Science  
It is not hard to see that the word problem for an HNN-extension (1) with A finite can be reduced in polynomial time to the word problem of the base group H.  ...  In this paper, we restrict to HNN-extensions (resp. amalgamated products) with finite associated (resp. amalgamated) subgroups, which is an important subcase.  ...  From the close relationship of HNN-extensions with amalgamated free products, a polynomial time reduction from the compressed problem for an amalgamated free product H 1 * H 2 | a = ϕ(a) (a ∈ A 1 ) (with  ... 
doi:10.1007/978-3-642-03351-3_23 fatcat:cdorifk36fdhfcbilz47rc3loa

Complexity of word problems for HNN-extensions [article]

Markus Lohrey
2021 arXiv   pre-print
It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H.  ...  The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory.  ...  The compressed word problem turns out to be useful for the solution of the word problem for an ascending HNN-extension: (3) in G.  ... 
arXiv:2107.01630v1 fatcat:dgv33bxrvjggpilmg2at7y3uhe

Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products

Markus Lohrey, Georg Zetzsche, Marc Herbstritt
2016 Symposium on Theoretical Aspects of Computer Science  
We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified  ...  This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem.  ...  HNN-extensions and amalgamated products.  ... 
doi:10.4230/lipics.stacs.2016.50 dblp:conf/stacs/LohreyZ16 fatcat:izbsudjb6fbkhg67szrcefx4ee

A relationship between HNN extensions and amalgamated free products in operator algebras [article]

Yoshimichi Ueda
2008 arXiv   pre-print
With a minor change made in the previous construction we observe that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product in both the von Neumann algebra  ...  We apply the observation to the questions of factoriality and type classification of HNN extensions of von Neumann algebras and also those of simplicity and K-theory of (reduced or universal) HNN extensions  ...  Firstly, with a minor change of the previous construction in [19] we see that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product.  ... 
arXiv:math/0601706v4 fatcat:ttc27f42rnfrhnbniydbhlrdri

Editorial

Andrei S. Morozov, Klaus Wagner
2011 Theory of Computing Systems  
Niko Haubold and Markus Lohrey determine the complexity levels of the compressed word problem for HNN-extensions and amalgamated free products.  ...  Artur Jeż and Alexander Okhotin find complexity levels for some algorithmic problems in conjunctive grammars over unary alphabet.  ... 
doi:10.1007/s00224-011-9328-5 fatcat:ihis5erdrzhw5hil74raavtrpi

Hilbert space compression for free products and HNN-extensions

Dennis Dreesen
2011 Journal of Functional Analysis  
We also investigate the case of HNN-extensions of a group relative to a subgroup which is finite or of finite index.  ...  Given the (equivariant or non-equivariant) Hilbert space compression of two groups, we find bounds on the compression of their free product.  ...  Acknowledgments I thank Alain Valette for reading previous versions of the article, for interesting conversations and for pointing out some open problems related to Hilbert space compression.  ... 
doi:10.1016/j.jfa.2011.08.012 fatcat:ibems3rosjeutg4z3je5a6am4q

Hilbert space compression for free products and HNN-extensions [article]

Dennis Dreesen
2010 arXiv   pre-print
We also investigate the Hilbert space compression of an HNN-extension of a group relative to a finite normal subgroup or a finite index subgroup.  ...  Given the Hilbert space compression of two groups, we find bounds on the Hilbert space compression of their free product.  ...  Acknowledgements I thank Alain Valette for reading previous versions of the article, for interesting conversations and for pointing out some open problems related to Hilbert space compression.  ... 
arXiv:1002.3879v3 fatcat:lhhr6hqvpffhdaarf2j7pav3ja

Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P [article]

Volker Diekert, Jürn Laun, Alexander Ushakov
2011 arXiv   pre-print
Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P.  ...  The main result is that the Word Problem of Higman's group is decidable in polynomial time.  ...  If this is the case, Π is not a power circuit and we stop with the output "no". From now on we assume ε(Q) ≥ 1.  ... 
arXiv:1103.1232v1 fatcat:rhrffduubnflppyu5dcjmkxovu

EFFICIENT ALGORITHMS FOR HIGHLY COMPRESSED DATA: THE WORD PROBLEM IN HIGMAN'S GROUP IS IN P

VOLKER DIEKERT, JÜRN LAUN, ALEXANDER USHAKOV
2012 International journal of algebra and computation  
Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is is decidable in polynomial time.  ...  The Word Problem of Higman's group is decidable in polynomial time. It is due to the last result that we were forced to advance the theory of power circuits. Thanks.  ...  The situation for G (2, 3) is open and related to questions in number theory. Baumslag and Higman groups are built via HNN extensions and amalgamated products.  ... 
doi:10.1142/s0218196712400085 fatcat:eau27rtkf5fwvmvsrhpcuusasi

Knapsack in graph groups, HNN-extensions and amalgamated products [article]

Markus Lohrey, Georg Zetzsche
2015 arXiv   pre-print
We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified  ...  This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem.  ...  This shows that the subset sum problem and the knapsack problem are NP-hard for the group F (Σ) × F (Σ), where for knapsack we allow integer exponents.  ... 
arXiv:1509.05957v1 fatcat:6232c63sdrhjxpfiunol2icshy

The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable

Alexei Myasnikov, Alexander Ushakov, Dong Wook Won
2011 Journal of Algebra  
We prove that the Word Problem in the Baumslag group G (1,2) = a, b; a a b = a 2 which has a non-elementary Dehn function is decidable in polynomial time.  ...  The compressed form of the Magnus breakdown algorithm We mentioned in Section 2.2 that the Magnus breakdown algorithm for the Word Problem in G (1, 2) represents the group G (1, 2) as the HNN extension  ...  This enables us to perform some variations of the standard algorithms in HNN extensions (or similar groups) keeping the actual rewriting in the compressed form.  ... 
doi:10.1016/j.jalgebra.2011.07.024 fatcat:zgerymcldrft7o5rre3uznfjpi

The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable [article]

Alexei Miasnikov, Alexander Ushakov, Dong Wook Won
2011 arXiv   pre-print
We prove that the Word problem in the Baumslag group G(1,2) which has a non-elementary Dehn function is decidable in polynomial time.  ...  This enables us to perform some variations of the standard algorithms in HNN extensions (or similar groups) keeping the actual rewriting in the compressed form.  ...  In section 2 we discuss algorithmic properties of elements of G (1, 2) as an HNN extension of the Baumslag-Solitar group, set up the notation, and outline the difficulty of solvig the Word problem using  ... 
arXiv:1102.2481v1 fatcat:4nhvsl47qjhypagvwv4ovpdxwe

Closure properties of knapsack semilinear groups [article]

Michael Figelius, Markus Lohrey, Georg Zetzsche
2019 arXiv   pre-print
, ..., x_k take values in the natural numbers): graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions.  ...  We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form g_1^x_1... g_k^x_k = g in a group G, where the variables x_1  ...  The first and second author were supported by the DFG research project LO 748/12-1.  ... 
arXiv:1911.12857v1 fatcat:graxo2mwuzhp7ja6latfxqufvi
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