Morphisms

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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“…See [9] for a general overview of morphisms. An involution θ : X −→ X is defined as a map such that θ 2 is the identity function.…”

Section: Formal Language Prerequisitesmentioning

confidence: 99%

On Hairpin-Free Words and Languages

Kari

Konstantinidis

Sosík

et al. 2005

Developments in Language Theory

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“…The main reason is that a direct product of two free monoids has a faithful representation in the multiplicative semigroup N 3×3 (which extends naturally that of a free monoid in the multiplicative semigroup of N 2×2 ). This allows us to encode Post Correspondence Problem and therefore to establish the undecidabibility of certain problems, see [16], also [11], [3] or [8]. E., g., the freeness of the subsemigroup of a finite number of matrices can be shown to be undecidable when one observes that the "uniquely decipherability property" in A * × B * is undecidable, [5].…”

Section: Preliminariesmentioning

confidence: 99%

Some decision problems on integer matrices

Choffrut

Karhumäki

2005

RAIRO-Theor. Inf. Appl.

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Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup.

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“…For undecidability, we show that there exists a regular set of trajectories S such that determining whether context-free languages are shp Σ (S, θ )-free for morphic or antimorphic involutions is impossible. The following results employ the undecidability of Post's Correspondence Problem (PCP); we refer the reader to Harju and Karhumäki [8] for an introduction. Theorem 6.2.…”

Section: Decidabilitymentioning

confidence: 99%

Hairpin Structures Defined by DNA Trajectories

Domaratzki

2006

DNA Computing

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We examine scattered hairpins, which are structures formed when a single strand of nucleotides folds into a partially hybridized stem and a loop. To specify different classes of hairpins, we use the concept of DNA trajectories, which allows precise descriptions of valid bonding patterns on the stem of the hairpin. DNA trajectories have previously been used to describe bonding between separate strands.We are interested in the mathematical properties of scattered hairpins described by DNA trajectories. We examine the complexity of the set of hairpin-free words described by a set of DNA trajectories. In particular, we consider the closure properties of language classes under sets of DNA trajectories of differing complexity. We address decidability of recognition problems for hairpin structures.

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Morphisms

Cited by 50 publications

References 57 publications

On Hairpin-Free Words and Languages

On Hairpin-Free Words and Languages

Some decision problems on integer matrices

Hairpin Structures Defined by DNA Trajectories

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