The Unambiguity of Segmented Morphisms

Freydenberger, Dominik D.; Reidenbach, Daniel

doi:10.1007/978-3-540-73208-2_19

Cited by 5 publications

(9 citation statements)

References 11 publications

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“…Definition 16 is derived from the research on the ambiguity of morphisms (cf. Freydenberger and Reidenbach [4]), where it is of major importance. Concerning the word v given in Example 15, we have S = {1, 3}, C = {5}, R = {2, 4} and N = ∅.…”

Section: The Imprimitive Hullmentioning

confidence: 99%

Morphically primitive words

Reidenbach¹,

Schneider²

2009

Theoretical Computer Science

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In the present paper, we introduce an alternative notion of the primitivity of words, that -unlike the standard understanding of this term -is not based on the power (and, hence, the concatenation) of words, but on morphisms. For any alphabet Σ, we call a word w ∈ Σ * morphically imprimitive provided that there are a shorter word v and morphisms h, h ′ : Σ * → Σ * satisfying h(v) = w and h ′ (w) = v, and we say that w is morphically primitive otherwise. We explain why this is a wellchosen terminology, we demonstrate that morphic (im-)primitivity of words is a vital attribute in many combinatorial domains based on finite words and morphisms, and we study a number of fundamental properties of the concepts under consideration.

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Section: The Imprimitive Hullmentioning

confidence: 99%

Morphically primitive words

Reidenbach¹,

Schneider²

2009

Theoretical Computer Science

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“…Note that α 0\2 is not the shortest known pattern in U (σ 0 ) \ U (σ 2 ). The similar pattern α 0\2 described by Freydenberger and Reidenbach [5] (which has a length of 53 and 15 different variables) is also contained in this set, but the present version allows the proof to be more concise.…”

Section: Morphisms With Less Than Three Segmentsmentioning

confidence: 75%

“…Reidenbach and Schneider [19]), we cannot refer to a computationally feasible method to successfully seek for any patterns in U (σ 1 ) \ U (σ 2 ), U (σ 0 ) \ U (σ 2 ) or U (σ 3 ) \ U (σ 2 ). In particular, this means that the patterns α 2 , α 0\2 and α 1\2 are "handmade" and therefore we cannot answer the question of whether there exist shorter examples than α 2 , the pattern α 0\2 from Freydenberger and Reidenbach [5] (cf. our remark below the proof of Theorem 17) and the modification of α 1\2 mentioned above that are suitable for proving Theorems 15, 17 and 18, respectively.…”

Section: Morphisms With Less Than Three Segmentsmentioning

confidence: 99%

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The unambiguity of segmented morphisms

Freydenberger

Reidenbach

2009

Discrete Applied Mathematics

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This paper studies the ambiguity of morphisms in free monoids. A morphism σ is said to be ambiguous with respect to a string α if there exists a morphism τ which differs from σ for a symbol occurring in α, but nevertheless satisfies τ (α) = σ(α); if there is no such τ then σ is called unambiguous. Motivated by the recent initial paper on the ambiguity of morphisms, we introduce the definition of a so-called segmented morphism σ n , which, for any n ∈ N, maps every symbol in an infinite alphabet onto a word that consists of n distinct factors in ab + a, where a and b are different letters. For every n, we consider the set U (σ n) of those finite strings over an infinite alphabet with respect to which σ n is unambiguous, and we comprehensively describe its relation to any U (σ m), m = n. Thus, our work features the first approach to a characterisation of sets of strings with respect to which certain fixed morphisms are unambiguous, and it leads to fairly counter-intuitive insights into the relations between such sets. Furthermore, it shows that, among the widely used homogeneous morphisms, most segmented morphisms are optimal in terms of being unambiguous for a preferably large set of strings. Finally, our paper yields several major improvements of crucial techniques previously used for research on the ambiguity of morphisms.

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“…The explicit detailed examination of the unambiguity of morphisms, as initiated by Freydenberger et al [4] and continued by Freydenberger and Reidenbach [3], offers a combinatorially rich theory and shows various cross-references to other topics in combinatorics on words, such as fixed points of morphisms, equality sets and the Post correspondence problem. This paper takes up some of the open problems in Freydenberger et al [4] -first of all the question, for which string, there exists an unambiguous erasing morphism.…”

Section: Introductionmentioning

confidence: 99%

Unambiguous Erasing Morphisms in Free Monoids

Schneider

2009

Lecture Notes in Computer Science

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This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ (α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601-628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.

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The Unambiguity of Segmented Morphisms

Cited by 5 publications

References 11 publications

Morphically primitive words

Morphically primitive words

The unambiguity of segmented morphisms

Unambiguous Erasing Morphisms in Free Monoids

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