The theorem of Büchi-Bruyère states that a subset of $N^d$ is $b$-recognizable if and only if it is $b$-definable. As a corollary, the first-order theory of $(N,+,V_b)$ is decidable (where $V_b(n)$ is the largest power of the base $b$ dividing $n$). This classical result is a powerful tool in order to show that many properties of $b$-automatic sequences are decidable. The first part of my lecture will be devoted to presenting this result and its applications to $b$-automatic sequences. Then I will move to $b$-regular sequences, which can be viewed as a generalization of $b$-automatic sequences to integer-valued sequences. I will explain bow first-order logic can be used to show that many enumeration problems of $b$-automatic sequences give rise to corresponding $b$-regular sequences. Finally, I will consider more general frameworks than integer bases and (try to) give a state of the art of the research in this domain.[-]

Après une introduction générale présentant les principaux modèles et problèmes étudiés, nous étudierons plus précisément trois sujets qui permettront d'illustrer des propriétés algorithmiques, des aspects algébriques et logiques de cette théorie :
- caractérisation, décision et minimisation des transducteurs séquentiels ;
- équivalence et fonctionnalité de transducteurs : de l'indécidabilité à la décidabilité ;
- présentation logique des transducteurs, et clôture par composition.[-]

Après une introduction générale présentant les principaux modèles et problèmes étudiés, nous étudierons plus précisément trois sujets qui permettront d'illustrer des propriétés algorithmiques, des aspects algébriques et logiques de cette théorie :
- caractérisation, décision et minimisation des transducteurs séquentiels ;
- équivalence et fonctionnalité de transducteurs : de l'indécidabilité à la décidabilité ;
- présentation logique des transducteurs, et clôture par composition.[-]

Graded modalities have been proposed in recent work on programming languages as a general framework for refining type systems with intensional properties. In particular, continuous endomaps of the discrete time scale, or time warps, can be used to quantify the growth of information in the course of program execution. Time warps form a complete residuated lattice, with the residuals playing an important role in potential programming applications. In this paper, we study the algebraic structure of time warps, and prove that their equational theory is decidable, a necessary condition for their use in real-world compilers. We also describe how our universal-algebraic proof technique lends itself to a constraint-based implementation, establishing a new link between universal algebra and verification technology.[-]

Subshifts of finite type are of high interest from a computational point of view, since they can be described by a finite amount of information - a set of forbidden patterns that defines the subshift - and thus decidability and algorithmic questions can be addressed. Given an SFT $X$, the simplest question one can formulate is the following: does $X$ contain a configuration? This is the so-called domino problem, or emptiness problem: for a given finitely presented group $0$, is there an algorithm that determines if the group $G$ is tilable with a finite set of tiles? In this lecture I will start with a presentation of two different proofs of the undecidability of the domino problem on $Z^2$. Then we will discuss the case of finitely generated groups. Finally, the emptiness problem for general subshifts will be tackled.[-]