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

2008
*
arXiv
*
pre-print

It is shown that the

arXiv:0811.3303v1
fatcat:ph4em5tysnfm3mqqqjm253gtjy
*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] . ...##
###
Compressed Word Problems in HNN-extensions and Amalgamated Products

2010
*
Theory of Computing Systems
*

It is not hard to see that the

doi:10.1007/s00224-010-9295-2
fatcat:bvn4l2vyvfdnblg3bdsedthnsy
*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 ...##
###
Compressed Word Problems in HNN-Extensions and Amalgamated Products
[chapter]

2009
*
Lecture Notes in Computer Science
*

It is not hard to see that the

doi:10.1007/978-3-642-03351-3_23
fatcat:cdorifk36fdhfcbilz47rc3loa
*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 ...##
###
Complexity of word problems for HNN-extensions
[article]

2021
*
arXiv
*
pre-print

It is shown that the

arXiv:2107.01630v1
fatcat:dgv33bxrvjggpilmg2at7y3uhe
*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. ...##
###
Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products

2016
*
Symposium on Theoretical Aspects of Computer Science
*

We also prove general transfer results: NP-membership of the knapsack

doi:10.4230/lipics.stacs.2016.50
dblp:conf/stacs/LohreyZ16
fatcat:izbsudjb6fbkhg67szrcefx4ee
*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*. ...##
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A relationship between HNN extensions and amalgamated free products in operator algebras
[article]

2008
*
arXiv
*
pre-print

With a minor change made

arXiv:math/0601706v4
fatcat:ttc27f42rnfrhnbniydbhlrdri
*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*. ...##
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Editorial

2011
*
Theory of Computing Systems
*

Niko Haubold

doi:10.1007/s00224-011-9328-5
fatcat:ihis5erdrzhw5hil74raavtrpi
*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. ...##
###
Hilbert space compression for free products and HNN-extensions

2011
*
Journal of Functional Analysis
*

We also investigate the case of

doi:10.1016/j.jfa.2011.08.012
fatcat:ibems3rosjeutg4z3je5a6am4q
*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*. ...##
###
Hilbert space compression for free products and HNN-extensions
[article]

2010
*
arXiv
*
pre-print

We also investigate the Hilbert space

arXiv:1002.3879v3
fatcat:lhhr6hqvpffhdaarf2j7pav3ja
*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*. ...##
###
Efficient algorithms for highly compressed data: The Word Problem in Higman's group is in P
[article]

2011
*
arXiv
*
pre-print

Using this new data structure it has been shown recently by Myasnikov, Ushakov

arXiv:1103.1232v1
fatcat:rhrffduubnflppyu5dcjmkxovu
*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. ...##
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EFFICIENT ALGORITHMS FOR HIGHLY COMPRESSED DATA: THE WORD PROBLEM IN HIGMAN'S GROUP IS IN P

2012
*
International journal of algebra and computation
*

Using this new data structure it has been shown recently by Myasnikov, Ushakov

doi:10.1142/s0218196712400085
fatcat:eau27rtkf5fwvmvsrhpcuusasi
*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*. ...##
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Knapsack in graph groups, HNN-extensions and amalgamated products
[article]

2015
*
arXiv
*
pre-print

We also prove general transfer results: NP-membership of the knapsack

arXiv:1509.05957v1
fatcat:6232c63sdrhjxpfiunol2icshy
*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. ...##
###
The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable

2011
*
Journal of Algebra
*

We prove that the

doi:10.1016/j.jalgebra.2011.07.024
fatcat:zgerymcldrft7o5rre3uznfjpi
*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. ...##
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The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable
[article]

2011
*
arXiv
*
pre-print

We prove that the

arXiv:1102.2481v1
fatcat:4nhvsl47qjhypagvwv4ovpdxwe
*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 ...##
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Closure properties of knapsack semilinear groups
[article]

2019
*
arXiv
*
pre-print

, ..., x_k take values

arXiv:1911.12857v1
fatcat:graxo2mwuzhp7ja6latfxqufvi
*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. ...
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