The mathematician who had little wisdom: a story and some mathematics

Cohen, Deb; Duncan, Andrew J.; Gilbert, Nyandwi; Howie, James

doi:10.1017/cbo9780511566073.006

Cited by 3 publications

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“…The inequality (7) holds if i = 0, i = 1 or m = 0. The inductive arguments for the above inequalities are then identical to the corresponding ones in Lemma 2.1 of [12].…”

Section: Lemma 21mentioning

confidence: 99%

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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 (In unpublished work I. Kapovich and Schupp showed it is solvable in exponential time [33].)another example.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 inequality (7) holds if i = 0, i = 1 or m = 0. The inductive arguments for the above inequalities are then identical to the corresponding ones in Lemma 2.1 of [12].…”

Section: Lemma 21mentioning

confidence: 99%

“…Cohen, Madlener & Otto built the first examples. in a series of papers [7,8,26] where Dehn functions were first defined. They designed their groups in such a way that the 'intrinsic' method of solving the word problem involves running a very slow algorithm which has been suitably 'embedded' in the presentation.…”

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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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“…These are not the only such examples (but we believe they are the first that are explicit and elementary): Cohen, Madlener and Otto [14,15,28] embedded algorithms (modular Turing machines, in fact) with running times like n → A k (n) in groups so that the running of the algorithm is displayed in van Kampen diagrams so as to make the Dehn function reflect the time-complexity of the algorithms. They state that their techniques produce yet more extreme examples as they also apply to an algorithm with running time like n → A n (n), and so yield a group with Dehn function that is recursive but not primitive recursive.…”

Section: Extreme Dehn Functions the Dehn Function Area(n) Of A Finitmentioning

confidence: 99%

Hydra groups

Dison¹,

Riley²

2013

Comment. Math. Helv.

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“…Here we want to emphasize that the computability is a very powerful tool in the theory of languages for interactions among different branches of the natural sciences (see [5,7,8,36]). In molecular biology, for example, the structure of DNA is done by means of sequences of nucleobases (Adenine, Thymine, Cytosine, Guanine) and it is possible to look at words on a 4-letters alphabet, for instance {A, T, G, C}, in which prescribed rules are assigned among the links of the 4 nucleobases.…”

Section: An Application Between Sequences Of Dna and Bachmentioning

confidence: 99%

“…These groups are defined in terms of generators and relations and the words which we obtain are on alphabets admitting DFAs with specific properties (synchronous bicombability, automaticity, biautomaticity or hyperbolicity). We refer to [7,8,13,22,28] for precise definitions. Here we want only emphasize the importance of Hydra groups by the next remark.…”

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confidence: 99%

Some considerations on Hydra groups and a new bound for the length of words

Otera

Russo

2014

Mathematica Slovaca

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ABSTRACT. After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins. Hydra groupsThe combinatorics of words is increased significatively in the last years, since it is a powerful tool for the description of several processes in pure and applied sciences. A classic reference on the topic is [24] and will be used for the basic notions. We recall that a nonempty set A (which will be always assumed to be finite) is said to be an alphabet and its elements are called letters. By A + we denote the free semigroup generated by A, i.e. the set of all finite sequences of letters with the operation of concatenation. Elements of A + will be called words. It is well known that each word can be written in a unique way as a sequence of letters, so it is possible to define for each word w its length |w|:It is useful (though not necessary) to introduce the formal empty word ε: that is a word of length 0 such that2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 57M50; Secondary 68R05, 68Q15. K e y w o r d s: isoperimetric functions, finitely presented groups, lengths of words, counterpoint, Myhill-Nerode's theorem.

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The mathematician who had little wisdom: a story and some mathematics

Cited by 3 publications

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Taming the hydra: The word problem and extreme integer compression

Taming the hydra: The word problem and extreme integer compression

Hydra groups

Some considerations on Hydra groups and a new bound for the length of words

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