Florence Levé

Quasiperiodic Sturmian words and morphisms

We characterize all quasiperiodic Sturmian words: A Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity.

On a conjecture about finite fixed points of morphisms

A conjecture of M. Billaud is: given a word w, if, for each letter x occurring in w, the word obtained by erasing all the occurrences of x in w is a fixed point of a nontrivial morphism fx, then w is also a fixed point of a non-trivial morphism. We prove that this conjecture is equivalent to a similar one on sets of words. Using this equivalence, we solve these conjectures in the particular case where each morphism fx has only one expansive letter.

Quasiperiodic and Lyndon episturmian words

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if, it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.

Directive words of episturmian words : equivalences and normalization

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.

Deterministic geoleader election in disoriented anonymous systems

Consider a network made of n nodes scattered in the 2-dimensional space. Nodes are anonymous and disoriented devices being unable to communicate. Anonymous refers to systems made of a priori indistinguishable nodes. By disoriented, we mean that the nodes share no kind of coordinate system nor common sense of direction. Such systems are typically wireless sensor networks or swarms of robots endowed with localization capabilities, for instance visibility sensors or pattern formation maps. We address the Geoleader Election (GE) problem which is to ensure that, solely based on their positions, the nodes deterministically agree on the same position of a single node, called the leader. We provide a complete characterization on the node positions, both for systems with common handedness (chirality) and for systems devoid of a common handedness. The characterization is based on a particular object from combinatorics on words, namely the Lyndon words.

Subject and Counter-Subject Detection for Analysis of the Well-Tempered Clavier Fugues

Fugue analysis is a challenging problem. We propose an algorithm that detects subjects and counter-subjects in a symbolic score where all the voices are separated, determining the precise ends and the occurrence positions of these patterns. The algorithm is based on a diatonic similarity between pitch intervals combined with a strict length matching for all notes, except for the first and the last one. On the 24 fugues of the first book of Bachs Well-Tempered Clavier, the algorithm predicts 66% of the subjects with a musically relevant end, and finally retrieves 85% of the subject occurrences, with almost no false positive.

Snap-Stabilizing PIF on Non-oriented Trees and Message Passing Model

Starting from any configuration, a snap-stabilizing protocol guarantees that the system always behaves according to its specification while a self-stabilizing protocol only guarantees that the system will behave according to its specification in a finite time. So, a snap-stabilizing protocol is a time optimal self-stabilizing protocol (because it stabilizes in 0 rounds). That property is very suitable in the case of systems that are prone to transient faults. There exist a lot of approaches of the concept of self-stabilization, but to our knowledge, snap-stabilization is the only variant of self-stabilization which has been proved power equivalent to self-stabilization in the context of the state model (a locally shared memory model) and for non anonymous systems. So the problem of the existence of snap-stabilizing solutions in the message passing model is a very crucial question from a practical point of view. In this paper, we present the first snap-stabilizing propagation of information with feedback (PIF) protocol for non-oriented trees in the message passing model. Moreover using slow and fast timers, the round complexity of our algorithm is in θ(h ×k) and θ((h ×k) + k 2), respectively, where h is the height of the tree and k is the maximal capacity of the channels. We conjecture that our algorithm is optimal.

Computational Analysis of Musical Form

Can a computer understand musical forms? Musical forms describe how a piece of music is structured. They explain how the sections work together through repetition, contrast, and variation: repetition brings unity, and variation brings interest. Learning how to hear, to analyse, to play, or even to write music in various forms is part of music education. In this chapter, we briefly review some theories of musical form, and discuss the challenges of computational analysis of musical form. We discuss two sets of problems, segmentation and form analysis. We present studies in music information retrieval (MIR) related to both problems. Thinking about codification and automatic analysis of musical forms will help the development of better MIR algorithms.

Computational fugue analysis

One of the pinnacles of form in classical Western music, the fugue is often used in the teaching of music analysis and composition. Fugues alternate between instances of a subject and other patterns and modulatory sections, called episodes. Musicological analyses are generally built on these patterns and sections. We have developed several algorithms to perform an automated analysis of a fugue, starting from a score in which all the voices are separated. By focusing on the diatonic similarities between pitch intervals, we detect subjects and countersubjects, as well as partial harmonic sequences inside the episodes. We also implemented tools to detect subject scale degrees, cadences, and pedals, as well as a method for segmenting the fugue into exposition and episodic parts. Our algorithms were tested on a corpus of 36 fugues by J. S. Bach and Dmitri Shostakovich. We provide formalized ground-truth data on this corpus as well as a dynamic visualization of the ground truth and of our computed results. The complete system showed acceptable or good results for about one half of the fugues tested, enabling us to depict their design.

On Quasiperiodic Morphisms

Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.