Compressed Word Problems in HNN-Extensions and Amalgamated Products

Haubold, Niko; Lohrey, Markus

doi:10.1007/978-3-642-03351-3_23

Cited by 4 publications

(7 citation statements)

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“…B be an isomorphism, and assume that the compressed word problem for G can be solved in polynomial time. Then, also the compressed word problem for the HNN-extension H D hG; tI t 1 at D '.a/.a 2 A/i can be solved in polynomial time [56].…”

Section: Theorem 27 ([10]) Let G Be a Finite Groupmentioning

confidence: 99%

“…If the compressed word problems of G and H can be solved in polynomial time, then also the compressed word problem for the amalgamated free product G A H can be solved in polynomial time [56].…”

Section: Theorem 27 ([10]) Let G Be a Finite Groupmentioning

confidence: 99%

“…This may lead to more efficient algorithms. Examples for this strategy can be found for instance in combinatorial group theory [56,57,94,96,124], computational topology [36,121,123], interprocedural analysis [51], and bisimulation checking [60,78].…”

Section: Introductionmentioning

confidence: 99%

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Algorithmics on SLP-compressed strings: A survey

Lohrey

2012

Groups - Complexity - Cryptology

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124

105

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Results on algorithmic problems on strings that are given in a compressed form via straight-line programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

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Section: Theorem 27 ([10]) Let G Be a Finite Groupmentioning

confidence: 99%

Section: Theorem 27 ([10]) Let G Be a Finite Groupmentioning

confidence: 99%

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Algorithmics on SLP-compressed strings: A survey

Lohrey

2012

Groups - Complexity - Cryptology

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124

105

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“…Proof. For (17), observe that the identity θ r (a m ) = θ r−1 (a m )θ r−1 (a m−1 ) and inducting on r gives that the words are equal in the free group. The words are identical because that on the right is positive (that is, contains no inverse letters) and so is freely reduced.…”

Section: Constraining Cancellationmentioning

confidence: 99%

“…Our methods also bear comparison with the work of Lohrey, Schleimer and their coauthors [17,18,21,22,23,24,32] on efficient computation in groups and monoids where words are given in compressed forms using straight-line programs and are compared and manipulated using polynomial-time algorithms due to Hagenah, Plandowski and Lohrey.…”

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confidence: 99%

Taming the hydra: The word problem and extreme integer compression

Dison

Einstein

Riley

2018

Int. J. Algebra Comput.

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For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison & Riley showed that a "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.and so it is possible to check efficiently when a word on a ±1 and b ±1 represents the identity by multiplying out the corresponding string of matrices.A celebrated 1-relator group due to Baumslag [1] provides a more dramatic example:Platonov [29] proved its Dehn function grows like n → ⌊log 2 n⌋ exp 2 ( exp 2 · · · (exp 2 (1)) · · · ), where exp 2 (n) := 2 n . (Earlier results in this direction are in [2,14,15].) Nevertheless, Miasnikov, Ushakov & Won [27] solve its word problem in polynomial time. (In unpublished work I. Kapovich and Schupp showed it is solvable in exponential time [33].)another example. Diekert, Laun & Ushakov [11] recently gave a polynomial time algorithm for its word problem and, citing a 2010 lecture of Bridson, claim it too has Dehn function growing like a tower of exponentials. The groups we focus on in this article are yet more extreme 'natural examples'. They arose in the study of hydra groups by Dison & Riley [12] . Let θ : F(a 1 , . . . , a k ) → F(a 1 , . . . , a k )be the automorphism of the free group of rank k such that θ(a 1 ) = a 1 and θ(a i ) = a i a i−1 for i = 2, . . . , k. The family G k := a 1 , . . . , a k , t | t −1 a i t = θ(a i ) ∀i > 1 , are called hydra groups. Take HNN-extensions Γ k := a 1 , . . . , a k , t, p | t −1 a i t = θ(a i ), [p, a i t] = 1 ∀i > 1 TAMING THE HYDRA 5 of G k where the stable letter p commutes with all elements of the subgroup H k := a 1 t, . . . , a k t .It is shown in [12] that for k = 1, 2, . . ., the subgroup H k is free of rank k and Γ k has Dehn function growing like n → A k (n). Here we prove that nevertheless:Theorem 2. For all k, the word problem of Γ k is solvable in polynomial time.(In fact, our algorithm halts within time bounded above by a polynomial of degree 3k 2 + k + 2-see Section 5.)1.3. The membership problem and subgroup distortion. Distortion is the root cause of the Dehn function of Γ k growing like n → A k (n). The massive gap between Dehn function and the time-complexity of the word problem for Γ k is attributable to a similarly massive gap between a distortion function and the time-complexity of a membership problem. Here are more details.Suppose H is a subgroup of a group G and G and H have finite generating sets S and T , respectively. So G has a word metric d S (g, h), the length of a shortest word on S ±1 representing g −1 h, and H has a word metric d T s...

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The Compressed Word Problem in HNN-Extensions

Lohrey

2014

The Compressed Word Problem for Groups

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The computational complexity of the word problem in HNNextension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. 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 main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups.

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Compressed Word Problems in HNN-Extensions and Amalgamated Products

Abstract: It is shown that the compressed word problem for an HNNextension H, t | t −1 at = ϕ(a)(a ∈ A) with A finite is polynomial time Turing-reducible to the compressed word problem for the base group H.

Cited by 4 publications

References 21 publications

Algorithmics on SLP-compressed strings: A survey

Algorithmics on SLP-compressed strings: A survey

Taming the hydra: The word problem and extreme integer compression

The Compressed Word Problem in HNN-Extensions

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