96 research outputs found
Decidability of the HD0L ultimate periodicity problem
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
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
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
Towards a statement of the S-adic conjecture through examples
The -adic conjecture claims that there exists a condition such that a
sequence has a sub-linear complexity if and only if it is an -adic sequence
satisfying Condition for some finite set of morphisms. We present an
overview of the factor complexity of -adic sequences and we give some
examples that either illustrate some interesting properties or that are
counter-examples to what could be believed to be "a good Condition ".Comment: 2
Deciding Properties of Automatic Sequences
In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results.
We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory.
Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words.
In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences.
We close with a chapter of open problems, drawn from the earlier chapters
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
Enumeration and Decidable Properties of Automatic Sequences
We show that various aspects of k-automatic sequences -- such as having an
unbordered factor of length n -- are both decidable and effectively enumerable.
As a consequence it follows that many related sequences are either k-automatic
or k-regular. These include many sequences previously studied in the
literature, such as the recurrence function, the appearance function, and the
repetitivity index. We also give some new characterizations of the class of
k-regular sequences. Many results extend to other sequences defined in terms of
Pisot numeration systems
- …