2,220 research outputs found

    A new proof for the decidability of D0L ultimate periodicity

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    We give a new proof for the decidability of the D0L ultimate periodicity problem based on the decidability of p-periodicity of morphic words adapted to the approach of Harju and Linna.Comment: In Proceedings WORDS 2011, arXiv:1108.341

    A Study of Pseudo-Periodic and Pseudo-Bordered Words for Functions Beyond Identity and Involution

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    Periodicity, primitivity and borderedness are some of the fundamental notions in combinatorics on words. Motivated by the Watson-Crick complementarity of DNA strands wherein a word (strand) over the DNA alphabet \{A, G, C, T\} and its Watson-Crick complement are informationally ``identical , these notions have been extended to consider pseudo-periodicity and pseudo-borderedness obtained by replacing the ``identity function with ``pseudo-identity functions (antimorphic involution in case of Watson-Crick complementarity). For a given alphabet Σ\Sigma, an antimorphic involution θ\theta is an antimorphism, i.e., θ(uv)=θ(v)θ(u)\theta(uv)=\theta(v) \theta(u) for all u,vΣu,v \in \Sigma^{*} and an involution, i.e., θ(θ(u))=u\theta(\theta(u))=u for all uΣu \in \Sigma^{*}. In this thesis, we continue the study of pseudo-periodic and pseudo-bordered words for pseudo-identity functions including involutions. To start with, we propose a binary word operation, θ\theta-catenation, that generates θ\theta-powers (pseudo-powers) of a word for any morphic or antimorphic involution θ\theta. We investigate various properties of this operation including closure properties of various classes of languages under it, and its connection with the previously defined notion of θ\theta-primitive words. A non-empty word uu is said to be θ\theta-bordered if there exists a non-empty word vv which is a prefix of uu while θ(v)\theta(v) is a suffix of uu. We investigate the properties of θ\theta-bordered (pseudo-bordered) and θ\theta-unbordered (pseudo-unbordered) words for pseudo-identity functions θ\theta with the property that θ\theta is either a morphism or an antimorphism with θn=I\theta^{n}=I, for a given n2n \geq 2, or θ\theta is a literal morphism or an antimorphism. Lastly, we initiate a new line of study by exploring the disjunctivity properties of sets of pseudo-bordered and pseudo-unbordered words and some other related languages for various pseudo-identity functions. In particular, we consider such properties for morphic involutions θ\theta and prove that, for any i2i \geq 2, the set of all words with exactly ii θ\theta-borders is disjunctive (under certain conditions)

    Decidability of the HD0L ultimate periodicity problem

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    In this paper we prove the decidability of the HD0L ultimate periodicity problem

    Monadic Second-Order Logic with Arbitrary Monadic Predicates

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    We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent characterizations. We consider the regularity question: given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates, and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer Science Journal version: ToCL'17, Transactions on Computational Logi

    On the complexity of algebraic number I. Expansions in integer bases

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    Let b2b \ge 2 be an integer. We prove that the bb-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion

    Inverse problems of symbolic dynamics

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    This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by substitutional system, and dynamical properties are considered as criteria of superword being generated by interval exchange transformation. As a consequence, one can get a morphic word appearing in interval exchange transformation such that frequencies of letters are algebraic numbers of an arbitrary degree. Concerning multydimensional systems, our main result is the following. Let P(n) be a polynomial, having an irrational coefficient of the highest degree. A word ww (w=(w_n), n\in \nit) consists of a sequence of first binary numbers of {P(n)}\{P(n)\} i.e. wn=[2{P(n)}]w_n=[2\{P(n)\}]. Denote the number of different subwords of ww of length kk by T(k)T(k) . \medskip {\bf Theorem.} {\it There exists a polynomial Q(k)Q(k), depending only on the power of the polynomial PP, such that T(k)=Q(k)T(k)=Q(k) for sufficiently great kk.
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