The entropy of a language is a measure of its complexity and a well-studied dynamical invariant. I consider two related questions: for a given class of languages, can this parameter be computed, and what values can it take?

In 1D tilings (subshifts) of finite type, we have known how to compute the entropy for 30 years, and the method gives an algebraic characterisation of possible values. In higher dimension, a surprise came in 2007: not only is the entropy not computable in general, but any upper-semi-computable real number appears as entropy - a weak computational condition. Since then new works have shown that entropy becomes computable again with aditionnal mixing hypotheses. We do not know yet where the border between computable and uncomputable lies.

In this talk, I will explore the case of general subshifts (not of finite type) in any dimension, hoping to shed some light on the finite type case. I relate the computational difficulty of computing the entropy to the difficulty of deciding if a word belongs to the language. I exhibit a threshold in the mixing rate where the difficulty of the problem jumps suddenly, the very phenomenon that is expected in the finite type case.

This is a joint work with Silvère Gangloff and Cristobal Rojas.

Joint work with Zoltán Ésik University of Szeged, Hungary

We equip the collection of labeled posets (partially ordered sets), abbreviated l.p., with different operations: series product (concatenation of l.p), parallel product (disjoint union of posets), omega-power (concatenation of an omega sequence of the same poset) and omega-product (concatenation of an omega sequence of possibly different posets, which has therefore infinite arity). We select four subsets of these operations and show that in each case the equational theory is axiomatizable. We characterize the free algebras in the corresponding varieties, both algebraically as classes which are closed under the above operations as well as combinatorially as classes of partially ordered subsets. We also study the decidability issues when the question makes sense.

Nous munissons la collection des posets étiquetés (ensembles partiellement), en abrégé p.e., de différentes opérations: lproduit série (concaténation de p.e.), produit parallèle (union disjointe de p.e.), omega puissance (concaténation d'une omega suite du même p.e.) et omega produit (concaténation d'une omega suite de p.e., éventuellement différents, donc d'arité infinie. Nous distinguons quatre sous-ensembles parmi les opérations ci-dessus et nous montrons que dans chaque cas la théorie équationnelle est axiomatisable. Nous caractérisons les algèbres libres dans les variétiés correspondante aussi bien algébriquement en tant classes d'algèbres fermées pour les opérations ci-dessus et combinatoriquement en tant que classes de structures ordonnées. Nous étudions aussi les problèmes de décidabilité quand ils ont un sens.

In a one-counter automaton (OCA), one can produce a letter from some finite alphabet, increment and decrement the counter by one, or compare it with constants up to some threshold. It is well-known that universality and language inclusion for OCAs are undecidable. In this paper, we consider OCAs with counter observability: Whenever the automaton produces a letter, it outputs the current counter value along with it. Hence, its language is now a set of words over an infinite alphabet. We show that universality and inclusion for that model are PSPACE-complete, thus no harder than the corresponding problems for finite automata. In fact, by establishing a link with visibly one-counter automata, we show that OCAs with counter observability are effectively determinizable and closed under all boolean operations.

In formal language theory, several different models characterize regular languages, such as finite automata, congruences of finite index, or monadic second-order logic (MSO). Moreover, several fragments of MSO have effective characterizations based on algebraic properties, the most famous example being the Schützenberger-McNaughton and Papert theorem linking first-order logic with aperiodic congruences. When we consider transducers instead of automata, such characterizations are much more challenging, because many of the properties of regular languages do not generalize to regular word functions. In this paper we consider functions that are definable by one-way transducers (rational functions). We show that the canonical bimachine of Reutenauer and Schützenberger preserves certain algebraic properties of rational functions, similar to the syntactic congruence for languages. In particular, we give an effective characterization of functions that can be defined by an aperiodic one-way transducer.

Program:

• (09h20 - 09h30) Opening
• (09h30 - 10h00) Serge Grigorieff : “Algorithmic randomness and uniform distribution modulo one”
• (10h00 - 10h30) Stéphane Demri : “Reasoning about data repetitions with counter systems”
• (10h30 - 11h00) Coffee Break
• (11h00 - 11h30) Michel Habib : “A nice graph problem coming from biology: the study of read networks”
• (11h30 - 12h00) Delia Kesner : “Completeness of Call-by-Need (A fresh view)”
• (12h00 - 12h30) Pierre Vial : “Infinite Intersection Types as Sequences: a New Answer to Klop's Problem”
• (12h30 - 14h00) Lunch (Buffon Restaurant - 17 rue Hélène Brion - Paris 13ème)
• (14h00 - 14h30) Verónica Becher : “Finite-state independence and normal sequences”
• (14h30 - 15h00) Brigitte Vallée : “Towards the random generation of arithmetical objects”
• (15h00 - 15h30) Valérie Berthé : “Dynamical systems and their trajectories”
• (15h30 - 16h00) Coffee Break
• (16h00 - 16h30) Nicolás Alvarez : “Incompressible sequences on subshifts of finite type”
• (16h30 - 17h00) Eugene Asarin : “Entropy Games”
• (17h00 - 18h00) Discussion about the future of LIA INFINIS

More details are available here.

Representations of braids as isotopy classes of laminations of punctured disks are related with a family of normal forms, which we call relaxation normal forms. Roughly speaking, every braid is identified with a picture on a punctured disk, and reducing step-by-step the complexity of this picture amounts to choosing a relaxation normal form of the braid.

We will study the right relaxation normal form, which belongs to this family of normal forms. We will show that it is regular, and that it is synchronously bi-automatic if and only if the braid group has 3 punctures or less.

A successful idea in the area of verification is to consider finite-state abstractions of infinite-state systems. A prominent example is the fact that many language classes satisfy a Parikh's theorem, i.e. for each language, there exists a finite automaton that accepts the same language up to the order of letters. Hence, provided that the abstraction preserves pertinent properties, this allows us to work with finite-state systems, which are much easier to handle.

While Parikh-style abstractions have been studied very intensely over the last decades, recent years have seen an increasing interest in abstractions based on the subword ordering. Examples include the set of (non necessarily contiguous) subwords of members of a language (the downward closure), or their superwords (the upward closure). Whereas it is well-known that these closures are regular for any language, it is often not obvious how to compute them. Another type of subword based abstractions are piecewise testable separators. Here, a separators acts as an abstraction of a pair of languages.

This talk will present approaches to computing closures, deciding separability by piecewise testable languages, and a (perhaps surprising) connection between these problems. If time permits, complexity issues will be discussed as well.

In this talk, we study a model computing relations over finite words, generalising one- and two-way transducers. The model, called two-way two-tape automaton, consists in a finite-state machine with two read-only tapes, each one with a reading head able to go both ways. We first emphasize its relation with 4-way automata, which recognize sets of two-dimensional arrays of letters called picture languages; such correspondence provides a proof of the undecidability of the model, and an example separating determinism and non-determinism. We then describe several techniques which, applied to our model, establish (non-)closure properties of the recognizable relations. Finally, the main result presented in this talk is that alternating two-way two-tape automata are not closed under complementation. The proof is a refinement of one of J. Kari for picture languages.

Joint work with Olivier Carton and Olivier Serre.

We study the classical problem of conjunctive query evaluation. This problem admits multiple formulations and has been studied in numerous contexts; for example, it is a formulation of the constraint satisfaction problem, as well as the problem of deciding if there is a homomorphism from one relational structure to another (which transparently generalizes the graph homomorphism problem).

We here restrict the problem according to the set of permissible queries; the particular formulation we work with is the relational homomorphism problem over a class of structures A, wherein each instance must be a pair of structures such that the first structure is an element of A. We present a comprehensive complexity classification of these problems, which strongly links graph-theoretic properties of A to the complexity of the corresponding homomorphism problem. In particular, we define a binary relation on graph classes and completely describe the resulting hierarchy given by this relation. This binary relation is defined in terms of a notion which we call graph deconstruction and which is a variant of the well-known notion of tree decomposition. We then use this graph hierarchy to infer a complexity hierarchy of homomorphism problems which is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized form of quantifier-free reductions. We obtain a significantly refined complexity classification of left-hand side restricted homomorphism problems, as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications, such as the classifications by Grohe-Schwentick-Segoufin (STOC 2001) and Grohe (FOCS 2003, JACM 2007).

After presenting this new advance, we will compare this line of research with another that aims to classify the complexity of the homomorphism problem where the second (target) structure is fixed, and that is currently being studied using universal-algebraic methods. We will also make some remarks on two intriguing variants, injective homomorphism (also called embedding) and surjective homomorphism.

This talk is mostly based on joint work with Moritz Müller that appeared in CSL-LICS ’14. In theory, the talk will be presented in a self-contained fashion, and will not assume prior knowledge of any of the studied notions.

9h30-9h45 welcome

9h45 Svetlana Puzynina 10h15 Sebastian Schoener 10h30 Célia Borlido 11h Thibault Godin 11h45 Benjamin Hellouin 12h15 Thomas Garrity

14h Olivier Carton 14h30 Sylvain Lombardy (LaBRI)– Démonstration du logiciel Vaucuson-R 15h30 Pablo Rotondo

Word equations are combinatorial equalities between strings of symbols, variables and functions, which can be used to model problems in a wide range of domains. While some complexity results for the solving of specific classes of equations are known, currently there does not exist any equation solver publicly available. Recently, we have proposed the implementation of such a solver based on Boolean satisfiability that leverages existing SAT solvers for this purpose. In this paper, we propose a new representation of equations on words having fixed length, by using an enriched graph data structure. We discuss the implementation as well as experimental results obtained on a sample of equations.

We consider deterministic automata which accept vectors of d integers for a fixed positive integer d. A deterministic automaton is then a finite representation of the sets of vectors it accepts. Many operations are particularly efficient with this representation, such as intersection of sets, testing whether two sets are equal or deciding whethersuch an automaton accepts a Presburger-definable set, that is a FO[+,<]-definable set over integers. We consider a similar problem for less expressive logics such as FO[<,0,moda] or \FO[+1,0,mod], where mod is the class of modular relations.

We state that it is decidable in time O(nlog(n)) whether a set of vectors accepted by a given finite deterministic automaton can be defined in the less expressive logic. The case of dimension 1 was already proven by Marsault and Sakarovitch. If the first algorithms gives a positive answer, the second one computes in time O(n^{3}log(n)) an existential formula in this logic that defines the same set. This improves the 2EXP time algorithm that can be easily obtained by combining the results of Leroux and Choffrut.

In this talk, it is intended to: -Introduce automata reading vectors of integers, -Present the logic FO[<,0,mod] over integers -Introduce classical tools relating automata to numbers. -Give an idea of how they can be applied to the above-mentionned problem.

We present in this work an alternative solution to perfect-information stochastic parity games. Instead of using the framework of μ-calculus, which hides completely the algorithmic aspect, we solve it by induction on the number of absorbing states.

We study an extension of FO^2[<], first-order logic interpreted in finite words in which only two variables are used. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables.

We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective algebraic characterization of the properties expressible in this logic. This enables us to prove, among many other things, that our new logic has strictly less expressive power than full first-order logic FO[<].

This is joint work with Andreas Krebs, Kamal Lodaya, and Paritosh Pandya, and will be presented at LICS2016.

This PhD is about two natural extensions of Finite Automata: the 2-way FA (2FA) and the 2-way transducers (2T).

The 2FA are computably equivalent to FA, even in their nondeterministic (2NFA) variant. However, in the Descriptional Complexity area, some questions remain. Raised by Sakoda and Sipser in 1978, the question of the cost of the simulation of 2NFA by 2DFA is still open. In this manuscript I give an answer in a restricted case in which the nondeterministic choices of the 2NFA may occur at the border of the input only (2ONFA). I show that every 2ONFA can be simulated by a 2DFA of subexponential (but superpolynomial) size. Under the assumptions L=NL, this cost is reduced to the polynomial level. Moreover, I prove that the complementation, and the simulation by a halting 2ONFA is polynomial.

Classical transducers (1-way) are well-known and admit nice characterizations (rational relations, logic). But their 2-way variant (2T) is still unknown, especially the nondeterministic case. In this area, my manuscript gives a new contribution: a algebraic characterization of the relations accepted by 2NT when both the input and output alphabets are unary. It can be reformulated as follows: each unary 2NT is equivalent to a sweeping (and even rotating) 2T. I also show that the assumptions made on the size of the alphabets are required.

The study of word relations, as algebraic object, and their transitive closure is another subject considered in my phd. When the relation belongs to some low level class, we are able to set the complexity of its transitive closure. This quickly becomes uncomputable when higher classes are considered.

Weighted automata (WA) extend finite-state automata defining functions from the set of words to a semiring S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine.

Regarding unambiguous WA over a group G, they can equivalently be described by a CRA whose registers take their values in G, and are updated by operations of the form X:=Y.c, with c in G and X,Y registers.

In this talk, I will give a characterisation of unambiguous weighted automata which are equivalent to cost register automata using at most k registers, for a given k. To this end, I will generalise two notions originally introduced by Choffrut for finite-state transducers: a twinning property and a bounded variation property, here parametrised by an integer k and that characterise WA/functions computing by a CRA using at most k registers.

This is a joint work with Pierre-Alain Reynier and Jean-Marc Talbot.

Matrices and matrix products play a crucial role in a representation and analysis of various computational processes. Unfortunately, many simply formulated and elementary problems for matrices are inherently difficult to solve even in dimension two, and most of these problems become undecidable in general starting from dimension three or four. Let us given a finite set of square matrices (known as a generator) which is forming a multiplicative semigroup S. The classical computational problems for matrix semigroups are:

• Membership (Decide whether a given matrix M belong to a semigroup S) and special cases such as: Identity (i.e if M is the identity matrix) and Mortality (i.e if M is the zero matrix) problems
• Vector reachability (Decide for a given vectors u and v whether exist a matrix M in S such that Mu=v)
• Scalar reachability (Decide for a given vectors u, v and a scalar L whether exist a matrix M in S such that uMv=L)
• Freeness (Decide whether every matrix product in S is unique, i.e. whether it is a code)

The undecidability proofs in matrix semigroups are mainly based on various techniques and methods for embedding universal computations into matrix products. The case of dimension two is the most intriguing since there is some evidence that if these problems are undecidable, then this cannot be proved using any previously known constructions. Due to a severe lack of methods and techniques the status of decision problems for 2×2 matrices (like membership, vector reachability, freeness) is remaining to be a long standing open problem. More recently, a new approach of translating numerical problems of 2×2 integer matrices into variety of combinatorial and computational problems on words and automata over group alphabet and studying their transformations as specific rewriting systems have led to a few results on decidability and complexity for some subclasses.

We introduce Sturmian colorings of regular trees, which are colorings of minimal unbounded factor complexity. Then, we classify Sturmian colorings into two families, namely cyclic and acyclic ones. We characterize acyclic Sturmian colorings in a way analogous to continued faction algorithm of Sturmian words. As for cyclic Sturmian colorings, we show that the coloring is a countable union of a periodic coloring, possibly union with a regular subtree colored with one color.

This is joint work with Seonhee Lim.

Une tuile de Wang est un carré dont les bords sont colorés. Étant donné un ensemble fini de tuiles de Wang, on cherche à savoir s'il est possible de paver le plan discret tout entier avec ces tuiles, en mettant une tuile par case de sorte que deux tuiles adjacentes aient la même couleur sur le bord qu'elles partagent. On s'intéresse plus particulièrement aux jeux de tuiles apériodiques, ceux pour lesquels un pavage existe, mais où il est impossible de paver le plan périodiquement. Ces jeux de tuiles sont une des briques de base de la majorité des résultats en dynamique symbolique multidimensionnelle.

Le premier jeu de tuiles apériodique trouvé par Berger avait 20426 tuiles, et le nombre de tuiles nécessaire a baissé progressivement jusqu'à ce que Culik obtienne en 1996 un jeu de 13 tuiles en utilisant une méthode due à Kari.

Avec Michael Rao, nous avons trouvé avec l'aide de plusieurs ordinateurs un jeu apériodique de 11 tuiles. Ce nombre est optimal : il n'existe pas de jeu apériodique de moins de 11 tuiles. Une des principales difficultés de cette recherche guidée par ordinateur est que nous cherchons une aiguille dans une botte de foin indécidable : il n'existe pas d'algorithme qui décide si un jeu de tuiles est apériodique.

Après une brève introduction au problème, je présenterai l'ensemble de 11 tuiles, ainsi que les techniques de théorie des automates et de systèmes de transitions qui ont permis de prouver (a) qu'il est apériodique, et (b) que c'est le plus petit.

An infinite language L determines an infinite word α if every string in L is a prefix of α. If L is regular, it is known that α must be ultimately periodic; conversely, every ultimately periodic word is determined by some regular language. We investigate other classes of languages to see what infinite words they determine, focusing on languages recognized by various kinds of automata.

Next, we consider prediction of infinite words by automata. In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the predictor's guesses are correct. We study the case in which the predictor is an automaton and the emitted values are drawn from a finite set; i.e., the emitted sequence is an infinite word.

The automata we consider are finite automata, pushdown automata, stack automata (a generalization of pushdown automata), and multihead finite automata, and we relate them to purely periodic words, ultimately periodic words, and multilinear words.

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People’s behaviors as small as possible, while Tribune wants to make it as large as possible. An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense. On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP ∩ coNP.

Joint work with Julien Cervelle, Aldric Degorre, Cătălin Dima, Florian Horn, and Victor Kozyakin.