The Compressed Word Problem for Groups

Lohrey, Markus

doi:10.1007/978-1-4939-0748-9

Cited by 52 publications

(84 citation statements)

References 116 publications

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“…Here we let val(A) = val(B)[i : j]. It is known [19] (see also [30]) that a given composition system can be transformed in polynomial time into an SLP with the same value. One can also allow mixed rules A → X 1 · · · X n where each X i is either a terminal, a nonterminal or an expression of the form B[i : j], which clearly can be eliminated in polynomial time.…”

Section: Reduction To Caterpillar Treesmentioning

confidence: 99%

Tree Compression Using String Grammars

et al. 2017

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We study the compressed representation of a ranked tree by a (string) straight-line program (SLP) for its preorder traversal, and compare it with the well-studied representation by straight-line context free tree grammars (which are also known as tree straight-line programs or TSLPs). Although SLPs turn out to be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height and Horton-Strahler number of a tree, tree navigation, or evaluation of Boolean expressions. Other problems on tree traversals turn out to be intractable, e.g. pattern matching and evaluation of tree automata.

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Section: Reduction To Caterpillar Treesmentioning

confidence: 99%

Tree Compression Using String Grammars

et al. 2017

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“…However, the operations that are needed in the algorithms, namely constructing the shortest word of a CFG and removing the prefix or suffix of a word, are in polynomial time using SLPs, cf. [11].…”

Section: Lemmamentioning

confidence: 99%

“…Instead of these uncompressed words nonterminals representing these words as SLPs are inserted in the transducers. All operations that are needed in the algorithms, namely constructing a SLP for the shortest word of an CFG, concatenation of SLPs, shifting the word produced by an SLP and removing the prefix or suffix of an SLP are in polynomial time [11].…”

Section: E Proof Of Theoremmentioning

confidence: 99%

Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time

Boiret¹,

Palenta²

2016

Developments in Language Theory

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We show that the equivalence of deterministic linear top-down treeto-word transducers is decidable in polynomial time. Linear tree-to-word transducers are non-copying but not necessarily order-preserving and can be used to express XML and other document transformations. The result is based on a partial normal form that provides a basic characterization of the languages produced by linear tree-to-word transducers.

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“…Together (22) and (23) tell us that ξ = 0. But then λ = 0 or µ = 0 because of the hypothesis a ǫ x i x a −ǫ x+1 i x in the instance of the a −1 2 and a 2 (which must be principal letters) in u each side of the a ξ 1 .…”

mentioning

confidence: 93%

“…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 for Groups

Cited by 52 publications

References 116 publications

Tree Compression Using String Grammars

Tree Compression Using String Grammars

Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time

Taming the hydra: The word problem and extreme integer compression

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