9 research outputs found
Unambiguous 1-Uniform Morphisms
A morphism h is unambiguous with respect to a word w if there is no other
morphism g that maps w to the same image as h. In the present paper we study
the question of whether, for any given word, there exists an unambiguous
1-uniform morphism, i.e., a morphism that maps every letter in the word to an
image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Unambiguous morphic images of strings
Motivated by the research on pattern languages, we study a fundamental combinatorial question on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence
of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism ρ with ρ ≠ σ and ρ(α) = σ(α). Our main result shows that a rich and natural class of strings is provided with unambiguous morphic images
Morphically primitive words
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 wΣ* 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 well-chosen 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
Morphic Primitivity and Alphabet Reductions
An alphabet reduction is a 1-uniform morphism that maps
a word to an image that contains a smaller number of dfferent letters.
In the present paper we investigate the effect of alphabet reductions on
morphically primitive words, i. e., words that are not a fixed point of
a nontrivial morphism. Our first main result answers a question on the
existence of unambiguous alphabet reductions for such words, and our
second main result establishes whether alphabet reductions can be given
that preserve morphic primitivity. In addition to this, we study Billaud's
Conjecture - which features a dfferent type of alphabet reduction, but
is otherwise closely related to the main subject of our paper - and prove
its correctness for a special case
Unambiguous 1-uniform morphisms
A morphism σ is unambiguous with respect to a word α if there is no other morphism τ that maps α to the same image as σ. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1
Unambiguous morphic images of strings
We study a fundamental combinatorial problem on morphisms in free semigroups: With
regard to any string α over some alphabet we ask for the existence of a morphism σ such
that σ(α) is unambiguous, i.e. there is no morphism T with T(i) ≠ σ(i) for some symbol
i in α and, nevertheless, T(α) = σ(α). As a consequence of its elementary nature, this
question shows a variety of connections to those topics in discrete mathematics which
are based on finite strings and morphisms such as pattern languages, equality sets and,
thus, the Post Correspondence Problem.
Our studies demonstrate that the existence of unambiguous morphic images essen-
tially depends on the structure of α: We introduce a partition of the set of all finite
strings into those that are decomposable (referred to as prolix) in a particular manner
and those that are indecomposable (called succinct). This partition, that is also known
to be of major importance for the research on pattern languages and on finite fixed
points of morphisms, allows to formulate our main result according to which a string α
can be mapped by an injective morphism onto an unambiguous image if and only if α is
succinct
Unambiguous Morphic Images of Strings
We study a fundamental combinatorial problem on morphisms in free semigroups: With
regard to any string α over some alphabet we ask for the existence of a morphism σ such
that σ(α) is unambiguous, i.e. there is no morphism T with T(i) ≠ σ(i) for some symbol
i in α and, nevertheless, T(α) = σ(α). As a consequence of its elementary nature, this
question shows a variety of connections to those topics in discrete mathematics which
are based on finite strings and morphisms such as pattern languages, equality sets and,
thus, the Post Correspondence Problem.
Our studies demonstrate that the existence of unambiguous morphic images essen-
tially depends on the structure of α: We introduce a partition of the set of all finite
strings into those that are decomposable (referred to as prolix) in a particular manner
and those that are indecomposable (called succinct). This partition, that is also known
to be of major importance for the research on pattern languages and on finite fixed
points of morphisms, allows to formulate our main result according to which a string α
can be mapped by an injective morphism onto an unambiguous image if and only if α is
succinct
Conditions on the existence of unambiguous morphisms
A morphism α is (strongly) unambiguous with respect to a word α if there is no
other morphism τ that maps α to the same image as σ. Moreover, α is said to be
weakly unambiguous with respect to a word α if σ is the only nonerasing morphism.....
On a conjecture about finite fixed points of morphisms
International audienceA conjecture of M. Billaud is: Given a word w, if, for each letter x occurring in w, the word obtained by erasing all the occurrences of x in w is a fixed point of a nontrivial morphism f_x, then w is also a fixed point of a nontrivial morphism. We prove that this conjecture is equivalent to a similar one on sets of words. Using this equivalence, we solve these conjectures in the particular case where each morphism f_x has only one expansive letter