2,220 research outputs found
A new proof for the decidability of D0L ultimate periodicity
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
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 , an antimorphic involution is an antimorphism, i.e., for all and an involution, i.e., for all . 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, -catenation, that generates -powers (pseudo-powers) of a word for any morphic or antimorphic involution . 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 -primitive words.
A non-empty word is said to be -bordered if there exists a non-empty word which is a prefix of while is a suffix of . We investigate the properties of -bordered (pseudo-bordered) and -unbordered (pseudo-unbordered) words for pseudo-identity functions with the property that is either a morphism or an antimorphism with , for a given , or 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 and prove that, for any , the set of all words with exactly -borders is disjunctive (under certain conditions)
Decidability of the HD0L ultimate periodicity problem
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
Monadic Second-Order Logic with Arbitrary Monadic Predicates
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
Let be an integer. We prove that the -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
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 (w=(w_n), n\in \nit) consists of a sequence of first binary numbers
of i.e. . Denote the number of different subwords
of of length by .
\medskip {\bf Theorem.} {\it There exists a polynomial , depending only
on the power of the polynomial , such that for sufficiently
great .
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