Emilie Charlier

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.

A DECISION PROBLEM FOR ULTIMATELY PERIODIC SETS IN NONSTANDARD NUMERATION SYSTEMS

Consider a nonstandard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.

Multidimensional generalized automatic sequences and shape-symmetric morphic words

An infinite word is S-automatic if, for all n>=0, its (n+1)th letter is the output of a deterministic automaton fed with the representation of n in the numeration system S. In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=1, we show that a multidimensional infinite word x:N^d->@S over a finite alphabet @S is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.

State complexity of testing divisibility

Under some mild assumptions, we study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of m N in the Fibonacci system is exactly 2m 2 .The 12th annual workshop, Descriptional Complexity of Formal Systems 2010, is taking place in Saskatoon, Canada, on August 8-10, 2010. It is jointly organized by the IFIP Working Group 1.2 on Descriptional Complexity and by the Department of Computer Science at the University of Saskatchewan. This volume contains the papers of the invited lectures and the accepted contributions.

Self-shuffling words

In this paper we introduce and study a new property of infinite words which is invariant under the action of a morphism: We say an infinite word

A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems

x\in \mathbb{A}^{\mathbb N},

Enumeration and decidable properties of automatic sequences

defined over a finite alphabet

Infinite self-shuffling words

\mathbb{A}

The growth function of S-recognizable sets

, is self-shuffling if x admits factorizations:

Multi-dimensional sets recognizable in all abstract numeration systems

x=\prod_{i=1}^\infty U_iV_i=\prod_{i=1}^\infty U_i=\prod_{i=1}^\infty V_i