Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for participants to get some hands-on training on how to use Walnut.[-]

This tutorial surveys computational aspects of cellular automata, a discrete dynamical model introduced by S. Ulam and J. von Neumann in the late 40s: a regular grid of finite state cells evolving synchronously according to a common local rule described by a finite automaton.

Formally, a cellular automaton is a tuple $(d, S, N, f)$ where $d \in \mathbb{N}$ is the dimension of the cellular space, $S$ is the finite set of states, $N \subseteq_{\text {finite }} \mathbb{Z}^d$ is the finite neighborhood and $f: S^N \rightarrow S$ is the local rule of the cellular automaton.

A configuration $c \in S^{\mathbb{Z}^d}$ is a coloring of the cellular space by states.

The global transition function $G: S^{\mathbb{Z}^d} \rightarrow S^{\mathbb{Z}^d}$ applies $f$ uniformly according to $N$, i.e. for every configuration $c \in S^{\mathbb{Z}^d}$ and every position $z \in \mathbb{Z}^d$ it holds
$$G(c)(z)=f\left(c\left(z+v_1\right), \ldots, c\left(z+v_m\right)\right) \quad \text { where } N=\left\{v_1, \ldots, v_m\right\} .$$
A space-time diagram $\Delta \in S^{\mathbb{Z}^d \times \mathbb{N}}$ is obtained by piling successive configurations of an orbit, i.e. for every time step $t \in \mathbb{N}$ it holds $\Delta_{t+1}=G\left(\Delta_t\right)$.

Computing inside the cellular space: The first part of the tutorial considers cellular automata as a universal model of computation. Several notions of universality are discussed: boolean circuit simulation, Turing universality, intrinsic universality. Special abilities of cellular automata as a model of massive parallelism are then investigated.

Computing properties of cellular automata: The second part of the tutorial considers properties of cellular automata and their computation. De Bruijn diagrams and associated regular languages are introduced as tools to decide injectivity and surjectivity of the global transition function in the one-dimensional case. Both immediate and dynamical properties are introduced, in particular the notion of limit set.

Computation and reduction: undecidability results: The last part of the tutorial considers computing by reduction to establish undecidability results on some properties of cellular automata: injectivity and surjectivity of the global transition function in higher dimensions, nilpotency and intrinsic universality in every dimension, a Rice's theorem for limit sets.[-]

Walnut is computer software, written in Java, that implements an algorithm to decide the truth of first-order logic statements in an extension of Presburger arithmetic known as Buchi arithmetic. It can be used to prove a wide variety of results in combinatorics on words and number theory. In this course we will give an introduction to the theory behind Walnut, examples of the types of results that can be proved with it, and exercises for participants to get some hands-on training on how to use Walnut.[-]

This tutorial surveys computational aspects of cellular automata, a discrete dynamical model introduced by S. Ulam and J. von Neumann in the late 40s: a regular grid of finite state cells evolving synchronously according to a common local rule described by a finite automaton.

Formally, a cellular automaton is a tuple $(d, S, N, f)$ where $d \in \mathbb{N}$ is the dimension of the cellular space, $S$ is the finite set of states, $N \subseteq_{\text {finite }} \mathbb{Z}^d$ is the finite neighborhood and $f: S^N \rightarrow S$ is the local rule of the cellular automaton.

A configuration $c \in S^{\mathbb{Z}^d}$ is a coloring of the cellular space by states.

The global transition function $G: S^{\mathbb{Z}^d} \rightarrow S^{\mathbb{Z}^d}$ applies $f$ uniformly according to $N$, i.e. for every configuration $c \in S^{\mathbb{Z}^d}$ and every position $z \in \mathbb{Z}^d$ it holds
$$G(c)(z)=f\left(c\left(z+v_1\right), \ldots, c\left(z+v_m\right)\right) \quad \text { where } N=\left\{v_1, \ldots, v_m\right\} .$$
A space-time diagram $\Delta \in S^{\mathbb{Z}^d \times \mathbb{N}}$ is obtained by piling successive configurations of an orbit, i.e. for every time step $t \in \mathbb{N}$ it holds $\Delta_{t+1}=G\left(\Delta_t\right)$.

Computing inside the cellular space: The first part of the tutorial considers cellular automata as a universal model of computation. Several notions of universality are discussed: boolean circuit simulation, Turing universality, intrinsic universality. Special abilities of cellular automata as a model of massive parallelism are then investigated.

Computing properties of cellular automata: The second part of the tutorial considers properties of cellular automata and their computation. De Bruijn diagrams and associated regular languages are introduced as tools to decide injectivity and surjectivity of the global transition function in the one-dimensional case. Both immediate and dynamical properties are introduced, in particular the notion of limit set.

Computation and reduction: undecidability results: The last part of the tutorial considers computing by reduction to establish undecidability results on some properties of cellular automata: injectivity and surjectivity of the global transition function in higher dimensions, nilpotency and intrinsic universality in every dimension, a Rice's theorem for limit sets.[-]

Tree Automata Completion is an algorithm computing, or approximating, terms reachable by a term rewriting system. For many classes of term rewriting systems whose set of reachable terms is known to be regular, this algorithm is exact. Besides, the same algorithm can handle **any** left-linear term rewriting system, in an approximated way, using equational 2 abstractions. Thanks to those two properties, we will see that regular languages and tree automata completion provide a promising alternative for automatic static analysis of functional programs.[-]

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk I will present the proof (joint with Hermiller, Holt and Rees) that the conjugacy growth series of a virtually cyclic group is rational, and then also confirm the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental (joint with Antolín). The result for non-elementary hyperbolic groups can be used to prove a formal language version of Rivin's conjecture for any finitely generated acylindrically hyperbolic group G, namely that no set of minimal length conjugacy representatives of G can be regular.[-]

Words $u$ and $v$ are defined to be $k$-abelian equivalent if every factor of length at most $k$ appears as many times in $u$ as in $v$. The $k$-abelian complexity function of an infinite word can then be defined so that it maps a number $n$ to the number of $k$-abelian equivalence classes of length-$n$ factors of the word. We consider some variations of extremal behavior of $k$-abelian complexity.

First, we look at minimal and maximal complexity. Studying minimal complexity leads to results on ultimately periodic and Sturmian words, similar to the results by Morse and Hedlund on the usual factor complexity. Maximal complexity is related to counting the number of equivalence classes. As a more complicated topic, we study the question of how much k-abelian complexity can fluctuate between fast growing and slowly growing values. These questions could naturally be asked also in a setting where we restrict our attention to some subclass of all words, like morphic words.[-]

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.[-]

The study of palindromes and their generalizations in a word has gained a lot of interest in the last 20 years, motivated by applications in physics, biology, discrete geometry, to name only a few. Using Sebastien Ferenczi as an example, we illustrate the computation of its palindromic complexity and its relation with the usual factor complexity, via an identity attributed to Brlek and Reutenauer involving also the palindromic defect. Periodic infinite words as well as the family of words with language closed by reversal also satisfy the identity. The identity remains valid when palindromic is replaced by $\sigma$-palindromic, and we also discuss some other patterns.[-]