Combinatorics of Words

Choffrut, Christian; Karhumäki, Juhani

doi:10.1007/978-3-642-59136-5_6

Cited by 245 publications

(147 citation statements)

References 98 publications

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“…For a general overview and fundamental results in combinatorics on words, the reader is referred to [5,13]. If w is an nonempty word, then ww is called a square and www is called a cube.…”

Section: Gc T At Cmentioning

confidence: 99%

“…Note: after careful inspection of the automaton in the proof of Proposition 6, one can derive that the actual size is at most 25 3 · 4 k + 14 3 and the number of states do not exceed 5 3 · 4 k − 2 3 . The above results indicate that the size of a minimal NFA for hp(τ, k) grows exponentially with k. However, one should recall that k is the minimal length of bond allowing for a stable hairpin.…”

Section: Definition 5 a Set Of Pairs Of Stringsmentioning

confidence: 99%

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On Hairpin-Free Words and Languages

Kari

Konstantinidis

Sosík

et al. 2005

Developments in Language Theory

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Abstract. The paper examines the concept of hairpin-free words motivated from the biocomputing and bioinformatics fields. Hairpin (-free) DNA structures have numerous applications to DNA computing and molecular genetics in general. A word is called hairpin-free if it cannot be written in the form xvyθ(v)z, with certain additional conditions, for an involution θ (a function θ with the property that θ 2 equals the identity function). We consider three involutions relevant to DNA computing: a) the mirror image function, b) the DNA complementarity function over the DNA alphabet {A, C, G, T } which associates A with T and C with G, and c) the Watson-Crick involution which is the composition of the previous two. We study elementary properties and finiteness of hairpin (-free) languages w.r.t. the involutions a) and c). Maximal length of hairpinfree words is also examined. Finally, descriptional complexity of maximal hairpin-free languages is determined.

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“…For a general overview and fundamental results in combinatorics on words, the reader is referred to [5,13]. If w is an nonempty word, then ww is called a square and www is called a cube.…”

Section: Gc T At Cmentioning

confidence: 99%

Section: Definition 5 a Set Of Pairs Of Stringsmentioning

confidence: 99%

On Hairpin-Free Words and Languages

Kari

Konstantinidis

Sosík

et al. 2005

Developments in Language Theory

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“…The conclusion is inescapable that even for such a fixed, well defined body of mathematical propositions, mathematical thinking is, and must remain, essentially creative." 13 It also gives support to the "quasi-empirical" view of mathematics, which sustains that although mathematics and physics are different, it is more a matter of degree than black and white [12,9]; see also [24].…”

Section: Discussionmentioning

confidence: 76%

Solving Problems with Finite Test Sets

Calude

Jürgensen

Legg

2000

Finite Versus Infinite

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Every finite and every co-finite set of non-negative integers is decidable. This is true and it is not, depending on whether the set is given constructively. A similar constraint is applicable in language theory and many other fields. The constraint is usually understood and, hence, omitted.The phenomenon of a set being finite, but possibly undecidable, is, of course, a consequence of allowing non-constructive arguments in proofs. In this note we discuss a few ramifications of this fact. We start out with showing that every number theoretic statement that can be expressed in first-order logic can be reduced to a finite set, to be called a test set. Thus, if one knew the test set, one could determine the truth of the statement. The crucial point is, of course, that we may not able to know what the finite test set is. Using problems in the class Π 1 of the arithmetic hierarchy as an example, we establish that the bound on the size of the test set is Turing-complete and that it is upper-bounded by the busy-beaver function.This re-enforces the fact that there is a vast difference between finiteness and constructive finiteness. In the context of the present re-opened discussion about the notion of computability -possibly extending its realm through new computational models derived from physics -the constraint of constructivity of the model itself may add another twist.

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“…Another area of research related to our work (and this was in fact our initial motivation for addressing the present problem) is the problem of covering a set of strings S with a set X of substrings in S, where X is said to cover S if every string in S can be written as a concatenation of the substrings in X [13,2] (see also [15] and [3] for a more general treatment of the combinatorial rank). Covering a set of strings S with a set X of substrings in S is indeed the Minimum Generating Set problem for unary alphabet under the unary encoding scheme.…”

Section: Introductionmentioning

confidence: 99%

On Finding Small 2-Generating Sets

Fagnot

Fertin

2009

Lecture Notes in Computer Science

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Abstract. Given a set of positive integers S, we consider the problem of finding a minimum cardinality set of positive integers X (called a minimum 2-generating set of S) such that every element of S is an element of X or is the the sum of two (non-necessarily distinct) elements of X. We give some elementary properties of 2-generating sets and prove that finding a minimum cardinality 2-generating set is hard to approximate within ratio 1 + ε for any ε > 0. We then prove our main result, which consists in a standard representation lemma for minimum cardinality 2-generating sets.

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Combinatorics of Words

Cited by 245 publications

References 98 publications

On Hairpin-Free Words and Languages

On Hairpin-Free Words and Languages

Solving Problems with Finite Test Sets

On Finding Small 2-Generating Sets

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