130 research outputs found
Decidability of the isomorphism and the factorization between minimal substitution subshifts
Classification is a central problem for dynamical systems, in particular for
families that arise in a wide range of topics, like substitution subshifts. It
is important to be able to distinguish whether two such subshifts are
isomorphic, but the existing invariants are not sufficient for this purpose. We
first show that given two minimal substitution subshifts, there exists a
computable constant such that any factor map between these subshifts (if
any) is the composition of a factor map with a radius smaller than and some
power of the shift map. Then we prove that it is decidable to check whether a
given sliding block code is a factor map between two prescribed minimal
substitution subshifts. As a consequence of these two results, we provide an
algorithm that, given two minimal substitution subshifts, decides whether one
is a factor of the other and, as a straightforward corollary, whether they are
isomorphic.Comment: 54 page
Decidability of the isomorphism and the factorization between minimal substitution subshifts
68 pagesClassification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic, but the existing invariants are not sufficient for this purpose. We first show that given two minimal substitution subshifts, there exists a computable constant R such that any factor map between these sub-shifts (if any) is the composition of a factor map with a radius smaller than R and some power of the shift map. Then we prove that it is decid-able to check whether a given sliding block code is a factor map between two prescribed minimal substitution subshifts. As a consequence of these two results, we provide an algorithm that, given two minimal substitution subshifts, decides whether one is a factor of the other and, as a straightforward corollary, whether they are isomorphic
Cobham-Semenov theorem and \NN^d-subshifts
We give a new proof of the Cobham's first theorem using ideas from symbolic
dynamics and of the Cobham-Semenov theorem (in the primitive case) using ideas
from tiling dynamics.Comment: 24 page
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
Automatic Sequences and Decidable Properties: Implementation and Applications
In 1912 Axel Thue sparked the study of combinatorics on words when he showed that the Thue-Morse sequence contains no overlaps, that is, factors of the form ayaya.
Since then many interesting properties of sequences began to be discovered and studied.
In this thesis, we consider a class of infinite sequences generated by automata, called the k-automatic sequences. In particular, we present a logical theory in which many properties of k-automatic sequences can be expressed as predicates and we show that such predicates are decidable.
Our main contribution is the implementation of a theorem prover capable of practically characterizing many commonly sought-after properties of k-automatic sequences. We showcase a panoply of results achieved using our method. We give new explicit descriptions of the recurrence and appearance functions of a list of well-known k-automatic sequences. We define a related function, called the condensation function, and give explicit descriptions for it as well. We re-affirm known results on the critical exponent of some sequences and determine it for others where it was previously unknown. On the more theoretical side, we show that the subword complexity p(n) of k-automatic sequences is k-synchronized, i.e., the language of pairs (n, p(n)) (expressed in base k) is accepted by an automaton. Furthermore, we prove that the Lyndon factorization of k-automatic sequences is also k-automatic and explicitly compute the factorization for several sequences. Finally, we show that while the number of unbordered factors of length n is not k-synchronized, it is k-regular
- …