Pôle Algorithmes et structures discrètes

Pôle Automates, structures et vérification

Pôle Preuves, programmes et systèmes

Équipe-projet INRIA $\pi r^2$

Équipe thématique Algorithmes et complexité

Équipe thématique Algèbre et calcul

Équipe thématique Automates et applications

Équipe thématique Combinatoire

Équipe-projet INRIA GANG

Équipe thématique Modélisation et vérification

Équipe thématique Preuves et programmes

Équipe thématique Systèmes complexes, réseaux, calcul distribué

Équipe thématique Théorie et algorithmique des graphes

## Séminaire des doctorants

#### Jour, heure et lieu

Le mercredi à 11h, salle 3052

Le calendrier des séances (format iCal).

Pour ajouter le calendrier des séances à votre agenda favori, souscrire au calendrier en indiquant ce lien

#### Contact(s)

Voir aussi le blog du séminaire, avec des résumés de certains exposés, à l'adresse : https://semidoc.github.io/.

Si vous êtes stagiaire, doctorant extérieur à l'IRIF, ou post-doc, et que vous souhaitez recevoir les annonces du séminaire doctorants, vous pouvez vous inscrire à la liste d'annonce : https://listes.univ-paris-diderot.fr/sympa/subscribe/semidoc-irif

### Séances passées

#### Année 2019

Séminaire des doctorants

Mercredi 6 mars 2019, 11 heures, Salle 3014

**Isaac Konan** (IRIF) *Partitions and Bijections*

I will discuss bijections for some well-known partition identities, such as Schur partition identity and q-binomial coefficient.

NB: Open talk, all you need is just how to count objects.

Séminaire des doctorants

Mercredi 27 février 2019, 11 heures, Salle 3052

**Pierre Ohlmann** (IRIF) *Lower bounds for arithmetic circuits via the Hankel matrix*

Our first and main conceptual result is a characterization result: we show that the size of the smallest circuit computing a given non-associative polynomial is exactly the rank of a matrix constructed from the polynomial and called the Hankel matrix. This result applies to the class of all circuits in both commutative and non-commutative settings, and can be seen as an extension of the seminal result of Nisan giving a similar characterization for non-commutative algebraic branching programs.

The study of the Hankel matrix provides a unifying approach for proving lower bounds for polynomials in the (classical) associative setting. We demonstrate this by giving alternative proofs of recent results proving superpolynomial and exponential lower bounds for different classes of circuits as corollaries of our characterization result.

Our main technical contribution is to provide generic lower bound theorems based on analyzing and decomposing the Hankel matrix. This yields significant improvements on lower bounds for circuits with many parse trees, in both (associative) commutative and non-commutative settings. In particular in the non-commutative setting we obtain a tight result showing superpolynomial lower bounds for any class of circuits which has a small defect in the exponent of the total number of parse trees.

#### Année 2018

Séminaire des doctorants

Mercredi 28 novembre 2018, 11 heures, Salle 3052

**Cédric Ho Thanh** (IRIF) *Type theoretical approach to opetopes*

Séminaire des doctorants

Mercredi 14 novembre 2018, 11 heures, Salle 3052

**Thomas Colcombet** (Automata team) *Writing (large) LaTeX documents with the knowledge package.*

Clearly, there are some pitfalls to avoid. Sometimes a scientifically excellent thesis turns out to be barely usable, because definitions are difficult to find, hard to parse, etc… And the reviewers get mad at you (but say it gently because they are polite). It is your duty to pay attention to all these details (it is also the duty of your PhD advisor to help you in that) and make a document as user friendly as possible.

One can find many documents describing how to write good science/a good thesis on the internet (read some of them!). I will not try to cover this wide subject. My goal in this talk will be to emphasize on some presentation points, and show you how some tools can help you in your writing (this is an advertisement for the LaTeX package « knowledge »).

If you want to test, you can bring your laptop with an up-to-date distribution of LaTeX.

Séminaire des doctorants

Mercredi 31 octobre 2018, 11 heures, Salle 3052

**Anupa Sunny & Zhouningxin Wang** *New PhD session*

by Anupa Sunny

Abstract: This talk is based on a paper by Amnon Ta-Shma on the construction of epsilon biased sets which have a support size close to the Gilbert-Varshamov bound, a notion from coding theory. We will look at the Rozenman-Wigderson construction of the epsilon-biased set in which the bias of a set is amplified by taking a walk over an expander graph. We will then look at Ta-Shma's construction which is based on a modified version of the zigzag product, namely the s-wide replacement product.

Homomorphism of signed graphs

by Zhouningxin Wang

Abstract: The signed graph is a graph whose edges are assigned with the signs + and -. A homomorphism of one graph to the other preserves the adjacencies and incidences of these two edges. We extend the concept of homomorphism for signed graphs. An intuitive example will be given to explain why we consider the homomorphism of signed graphs. We will give the minimum signed graph, namely Spal_5, to which all the signed K_4-minor-free graph admits homomorphism to. In the last part, we will show the necessary and sufficient conditions for a signed K_4-subdivision being a core.

Séminaire des doctorants

Mercredi 17 octobre 2018, 11 heures, Salle 3052

**Abishek De & Simon Mauras** *Newcomers' session*

The distributed synthesis problem is about constructing correct distributed systems, i.e., systems that satisfy a given specification. We consider a slightly more general problem of distributed control, where the goal is to restrict the behavior of a given distributed system in order to satisfy the specification. Our systems are finite state machines that communicate via rendezvous (asynchronous automata). There are a few classes of systems for which the problem has been shown decidable. We solve it for free choice systems, systems whose entire behaviour can be expressed in a (possibly infinite) tree.

Simon Mauras : Social choice theory, and a small survey about rank aggregation

How should we vote? This question has been adressed by philosophers and mathematicians since the XVIIIth century, but no satisfactory solution exists. The talk will start with classical results on social choice theory and move on to the aggregation of rankings seen as an optimization problem. We will discuss NP-Hardness, Hardness of approximation and Approximation algorithms for several variants of this problem.

Séminaire des doctorants

Mercredi 12 septembre 2018, 11 heures, Salle 3052

**Farzad Jafarrahmani** *Denotational semantics of Linear Logic with least and greatest fixpoints*

Short session

Séminaire des doctorants

Mercredi 27 juin 2018, 11 heures, Salle 3052

**Victor Lanvin** (Équipes Preuves, programmes, et Systèmes) *Introduction to gradual typing with union and intersection types*

Note: this will (should) be a very general presentation about gradual typing and set-theoretic types consisting mostly of practical examples and without too many technical details. Don't hesitate to bring your computer, a book, or your Nintendo Switch™ if you already know the topic.

Séminaire des doctorants

Mercredi 20 juin 2018, 11 heures, Salle 3052

**Laurent Feuilloley** (Équipes Compsys, GANG et graphes) *Distributed decision*

The underlying model of this study is the local model. The local model is defined to answer questions of the following type: given a communication network, whose nodes are machines, and edges are communication links, is it possible that the nodes solve some task X, if they communicate only with the nodes that are close to them? A classic problem is colouring: can a node choose a colour, with only the knowledge of a small neighbourhood of the graph, such that the colours chosen by the nodes form a proper colouring of the graph? As in the centralized setting, it is interesting to study decision problems, that are yes-no questions, and to define complexity classes to classify these problems; this is distributed decision.

The complexity class we use as the equivalent of the class P in the centralized setting, is pretty small, and it is then natural to look at some form of non-determinism, to have a chance to understand more problems. In this model, non-determinism can be thought as a piece of global information that can be verified locally. The theoretical motivation is that to understand how local a problem is, one can ask how much global information is needed to solve it. The more practical motivation is that if one can design schemes with little global information then it can help to design more robust distributed algorithms such as self-stabilizing algorithms. The results I will present play with different natural notions of non-determinism, and how they influence the complexity classes defined.

I will spend time to carefully describe the model, thus no prior knowledge is needed.

Séminaire des doctorants

Mercredi 6 juin 2018, 11 heures, Salle 3052

**Pierre Ohlmann & Sidi Mohammed Beillahi** (Équipe automate & Équipe vérification) *Unifying non-commutative arithmetic circuit lower bounds & Robustness of Programs Against Consistency Relaxation*

We develop an algebraic lower bound technique in the context of non-commutative arithmetic circuits. To this end, we introduce polynomials for which the multiplication is also non-associative, and focus on their circuit complexity. We show a connection with multiplicity tree automata, leading to a general algebraic characterization. We use it to derive meta-theorems for the non-commutative case, and highlight numerous consequences in terms of lower bounds.

&&

Robustness of Programs Against Consistency Relaxation (Sidi Mohammed Beillahi)

Sequential Consistency (SC) and Serializability (Ser) are the classical consistency models for concurrent and distributed programs. They are the intuitive models for developers. Due to the costly synchronization required by the two models, most of existing memory models and distributed implementations of data structures do not use these two models. Instead, in order to reduce the latency and remove unnecessary synchronization, modern processors and databases adopt relaxed and weaker consistency models. However, this weakening of the consistency models implies new unexpected behaviors when running programs over the weaker models. We address in this work the problem of detecting unexpected behaviors of a program that were observed when weakening the consistency model. In particular, we check whether the two sets of executions traces of a program over the SC (resp, Ser) model and some weaker consistency model coincide or not. We characterize the set of executions traces that violate this equality and drive a decision procedure to detect these traces. In the case where there are no traces that violate this equality we refer to a program to be Robust.

A joint work with Ahmed Bouajjani and Constantin Enea

Séminaire des doctorants

Mercredi 23 mai 2018, 11 heures, Salle 3052

**Léo Stefanesco** (Algebra and calculus, proofs and programs teams) *An Asynchronous Soundness Theorem for Concurrent Separation Logic*

—

Abstract:

Concurrent separation logic (CSL) is a specification logic for concurrent imperative programs with shared memory and locks. In this talk, we develop a concurrent and interactive account of the logic inspired by asynchronous game semantics. To every program C, we associate a pair of asynchronous transition systems [C]S and [C]L which describe the operational behavior of the Code when confronted to its Environment, both at the level of machine states (S) and of machine instructions and locks (L). We then establish that every derivation tree π of a judgment Γ ⊢ {P}C{Q} defines a winning and asynchronous strategy [π] with respect to both asynchronous semantics [C]S and [C]L. From this, we deduce an asynchronous soundness theorem for CSL, which states that the canonical map L : [C]S → [C]L from the stateful semantics [C]S to the stateless semantics [C]L satisfies a basic fibrational property. We advocate that this fibrational property provide a clean and conceptual explanation for the usual soundness theorem of CSL, including the absence of data races.

(Joint work with Paul-André Melliès)

Séminaire des doctorants

Mercredi 2 mai 2018, 11 heures, Salle 3052

**Emiliano Lancini** (Laboratoire d'Informatique de Paris Nord) *Box-Total Dual Integrality and k-Edge-Connectivity*

Nowadays it is often required that telecommunication networks keep unaltered their functionality even after the failure of some of their links. This leads to a generalisation of the minimum spanning tree problem named k-edge-connected spanning subgraph problem.

In this talk we show a characterisation of a graph class in terms of integrality properties of polyhedra naturally associated to the k-edge-connected spanning subgraph problem.

The concept of total dual integrality (TDI) dates back to the works of Edmonds, Giles and Pulleyblank in the late 70's, and is strongly connected to min-max relations in combinatorial optimisation.

The system Ax>=b is TDI if, in the following equation, for each integer vector c, for which the minimum in the following equation is finite, there exists an integer optimal solution for the maximum.

min {cx: Ax>= b} = max {yb: yA = c, y >= 0}

We are interested in the stronger property of box-TDIness. A system Ax>=b is called *box-TDI* if the system Ax >= b, l ⇐ x ⇐ u is TDI for all rational vectors l and u.

We prove that, for k>=2, the k-edge-connected spanning subgraph polyhedron is a box-TDI polyhedron if and only if the graph is series-parallel. Moreover, in this case, we provide a box-TDI system with integer coefficients describing this polyhedron.

Séminaire des doctorants

Mercredi 25 avril 2018, 11 heures, Salle 3052

**Raphaëlle Crubillé** (Algebra and calculus, proofs and programs teams) *Probabilistic Stable Functions on Discrete Cones are Power Series.*

Séminaire des doctorants

Mercredi 18 avril 2018, 11 heures, Salle 3052

**Paulina Cecchi & Antoine Allioux** (Automata, Combinatorics teams & Algebra and calculus, proofs and programs teams) *New PhD student introduction session*

Title: Some interactions between words combinatorics and symbolic dynamics.

Abstract: Word combinatorics has been fruitfully used to study several topological and mesure-theoretic properties of dynamical systems, through the analysis of suitably chosen symbolic dynamical systems. In this talk, we will introduce some notions of symbolic dynamics and present some examples which illustrate how word combinatorics can be used as a tool to solve questions arising from this branch of mathematics.

*

* Antoine Allioux

Title: The curse of Martin-Löf identity type

Abstract: The identity type of Intuitionistic Type Theory (ITT) endows types with a structure of infinity-groupoid. This higher structure follows from the fact that the Uniqueness of Identity Proof (UIP) is not derivable in ITT. Homotopy Type Theory (HoTT) takes advantage of this observation by enriching ITT with new principles which are coherent with this interpretation, namely the Univalence Axiom and the Higher Inductive Types (HITs).

HITs are a generalization of inductive types which allow, in addition to create regular inhabitants of an inductive type, to postulate identities between them as well as identities between these identities, and that ad infinitum. It is then tempting to try to present mathematical structures using these new types like one would do in mathematics using generators and relations.

However, problems quickly arise as soon as one needs a strict equality. Indeed, the identity type expresses a weak equality leading to the usual coherence problems. Trying to solve these naively, we run into the problem of having to define an infinite sequence of coherence data.

If HoTT is to be a credible foundation of mathematics, the question of formalizing structures which need a strict equality is crucial. The answer to this question rests, in part, upon our achievement to either present these structures differently in the existing theory or to enrich it so that it becomes tractable to express them.

*

Séminaire des doctorants

Mercredi 11 avril 2018, 11 heures, Salle 3052

**Brieuc Guinard** *Intermittent Locomotion in Graphs*

Séminaire des doctorants

Mercredi 28 mars 2018, 11 heures, Salle 3052

**Yann Hamdaoui** (Proofs and Programs and Conception and Analysis of Systems teams) *Translating a Concurrent Lambda Calculus into Linear Logic proof (nets)*

Séminaire des doctorants

Mercredi 21 mars 2018, 11 heures, Salle 3052

**Mengchuan Zou** (Theory and algorithmics of graphs team) *Generalization of binary search in trees and other structures*

We will also introduce some known facts on other structures and how tree search problem is related to other problems via equivalences.

Séminaire des doctorants

Mercredi 21 février 2018, 11 heures, Salle 3052

**Zeinab Galal & Ranadeep Biswas** (Algebra and computation & Modeling and verification) *Species of structure: a Bridge between Differential Lambda Calculus and Combinatorics & Verifying Database Histories*

Species of structure lie at the intersection of combinatorics and denotational semantics. They were first introduced by Joyal as a unified framework for the theory of generating series in enumerative combinatorics and multiple tools were developed for the resolution of differential equations of species in this setting. Later, Fiore presented a generalized definition that both encompasses Joyal's species and constitutes a model of linear logic.

We will first introduce and connect the different viewpoints of species of structure and their series counterpart (analytic and normal functors) presented by Joyal, Girard and Hasegawa. Next, we will examine how the bicategory of generalized species of structure forms a model of differential linear logic.

As our end goal is to develop methods of resolution of differential equations for λ-terms, we will investigate the possible directions to tackle this problem.

&

*Verifying Database Histories*

Popular databases offer control over the isolation level to which the operations in one transaction are visible to the operations in other concurrent transactions. Lower levels allow weaker consistency. So, we have to ensure that the histories of a database are consistent with their isolation levels.

Unfortunately, these isolation levels are mostly defined as low-level operations which makes it complicated to reason about the behavior of the system running under those isolations.

In this talk, we will present some popular isolation levels and consistency criteria for databases. We will introduce a framework, in which it becomes easier to formally reason about the behavior of a system. Then we will explore the complexities of deciding some consistency criteria using that framework.

Séminaire des doctorants

Mercredi 14 février 2018, 11 heures, Salle 3052

**Narcisse Nya Kamtchoum** (LIP6) *Modèles analytiques pour les performances des réseaux cellulaires*

Cependant, les opérateurs ont du mal à s'adapter à la proportion toujours grandissante d'utilisateurs mobiles et à leur offrir une qualité d'expérience (QoE) satisfaisante. Dans ce contexte, il est importante pour les opérateurs et les équipementiers de disposer d'outils simples et efficaces pour mieux comprendre le comportement de leurs réseaux et évaluer la qualité des services offerts aux utilisateurs. Notre objectif est de proposer des modèles analytiques pour l'évaluation des performances des réseaux cellulaires en tenant compte de la mobilité des utilisateurs. Tout en permettant de résoudre des problèmes d'évaluation de performance les plus complexes, ces modèles se doivent d'être simple afin de faciliter leur utilisation.

Le séminaire sera donné en français.

Séminaire des doctorants

Mercredi 7 février 2018, 11 heures, Salle 3052

**Nicolas Jeannerod** (Team Analysis and conception of systems) *Unix filesystems and First-Order Theory of an Algebra of Feature Trees with Updates*

We investigate a logic of an algebra of trees with an update operation, which expresses that a tree is obtained from an input tree by replacing a particular direct subtree while leaving the rest intact. This operation improves on the expressivity of existing logics of tree algebras in our case of feature trees. These allow for an unbounded number of children of a node in a tree.

We show an efficient way of testing the satisfiability of existential clauses in this algebra that can lead to an efficient implementation of our symbolic execution engine. We also show the decidability of the first-order theory of this algebra via a weak quantifier elimination procedure which is allowed to swap existential quantifiers for universal quantifiers.

Séminaire des doctorants

Mercredi 31 janvier 2018, 11 heures, Salle 3052

**Thomas Williams** (Gallium, INRIA) *Refactoring ML programs using ornaments*

In this talk, I will explain how ornaments can be used to automatically lift a function. I will present a prototype implementation of lifting along ornaments for a subset of OCaml and describe some families of use cases. I will introduce a principled approach to obtaining a lifting from the base code, as abstraction followed by specialization. I will explain how this approach allows us to prove the correctness of the lifting.

Séminaire des doctorants

Mercredi 24 janvier 2018, 11 heures, Salle 3052

**Alessandro Luongo & Ny Aina Andriambolamalala** (Algorithms and complexity team & Combinatorics team) *Recent updates in quantum machine learning & Election de leader dans un réseau radio simple saut avec detection de collision*

In this talk we are going through some recent algorithms in the field of quantum machine learning. Most of the techniques use tools from quantum algorithmics such as counting, optimizing, estimating distances and singular values which will be introduced here. Using these primitives it's possible to build more complex operations of a matrix algebra. I'll also describe a classical machine learning algoritm in the process of being translated in a fully fledged quantum algorithm. This is the first biologically plausible quantum algorithm with an exponential speedup w.r.t the dimension of the space and the number of datapoints. This quantum algorithm has been simulated and used to classify handwritten digits with high accuracy.

*Election de leader dans un réseau radio simple saut avec detection de collision*

Les résultats de Dan Willard (1986) montrent un algorithme randomizé d'élection de leader en temps moyen $O(\log\log{n})$.

Depuis, la question de savoir s'il existe un algorithme convergeant en temps log-logarithmique mais avec très forte probabilité est ouverte.

Nous répondons affirmativement à cette question. Nous montrons aussi comment utiliser nos résultats pour élaborer des protocoles d'élection dans divers modèles de systèmes distribués.

These are two newcomers talk, 30 minutes each. The first will be in English, the second in French.

#### Année 2017

Séminaire des doctorants

Mercredi 20 décembre 2017, 11 heures, Salle 3052

**Leo Stefanesco** (Équipes “Preuves et Programmes” et “Algèbre et Calcul”) *“A concise introduction to logical relations” followed by “A Logical Relation for Monadic Encapsulation of State”*

Logical relations are a powerful technique to prove properties about programs. In particular, for proving that two programs are contextually equivalent.

In this talk, we will see that, in System F (aka the polymorphic lambda calculus), the only program of type ∀ a, a → a is the identity.

I will also sketch how to extend logical relations to realistic languages such as ML.

2nd part (POPL talk rehearsal):

A Logical Relation for Monadic Encapsulation of State

We present a logical relations model of a higher-order functional programming language with impredicative polymorphism, recursive types, and a Haskell-style ST monad type with runST. We use our logical relations model to show that runST provides proper encapsulation of state, by showing that effectful computations encapsulated by runST are heap independent. Furthermore, we show that contextual refinements and equivalences that are expected to hold for pure computations do indeed hold in the presence of runST. This is the first time such relational results have been proven for a language with monadic encapsulation of state. We have formalized all the technical development and results in Coq.

Séminaire des doctorants

Mercredi 13 décembre 2017, 11 heures, Salle 3052

**Simon Halfon** (ENS Cachan) *Well Quasi-Orders and Extreme Stratospheric-Complexity-Classes of Death — Beaux pré-ordres et classes de complexité stratosphériques de la mort*

No special knowledge is required to follow the talk.

—

Nous savons tous qu’il est très pratique pour prouver la terminaison d’un algorithme d’utiliser des ordres bien fondés. Cependant, il est commun de penser que cette technique ne donne aucune information sur la complexité de l’algorithme, car la preuve est non constructive. Dans cet exposé, je présenterai quelques idées pour extraire une borne supérieur de complexité d’une telle preuve de terminaison. J’essaierai de vous donner une idée de la combinatoire compliquée que cela engendre, et qui résulte en des complexité très élevée. Bouteilles d’oxygène et combinaisons pressurisées obligatoire, on s’envole au delà de l’Élémentaire.

Aucune connaissance spécifique n'est nécessaire pour suivre l'exposé.

Séminaire des doctorants

Mercredi 6 décembre 2017, 11 heures, Salle 3052

**Axel Osmond & Yassine Hamoudi** ([1] “Algebra and calculus” and “Proofs and programs” teams, [2] Algorithms and complexity team) *New PhD student introduction session. [1] From pointless topology to formal topology [2] ACC0 and multiparty communication: fighting the log n barrier.*

Formal topology goes further into this last direction, using systems of axioms about coverings as a deductive systems which leads to a type-theoretic flavored, predicative and constructive topology, endowed with multiple and finer notions of points, separability… and suited for intuitionistic reasoning.

[2] The Number On the Forehead model is a multiparty communication game between k players that collaboratively want to evaluate a given function F : X1 x … x Xk → Y on some input (x1, …, xk) by broadcasting bits according to a predetermined protocol. The input is distributed between the players in such a way that each player i sees all of it except xi (as if xi is written on the forehead of player i).

A central open question in this model, called the log n barrier, is to find a function which is hard to compute when the number of players is polylog(n) or more (where the xi's have size poly(n)). This has an important application in circuit complexity, as it could help to separate ACC0 from other complexity classes.

In this talk, we will recall first the connection between ACC0 and communication complexity, and then describe a new efficient communication protocol that prevents some important functions from breaking the log n barrier.

Séminaire des doctorants

Mercredi 22 novembre 2017, 11 heures, Salle 3052

**Jules Chouquet** (Proofs and Programs team) *Linear logic proof nets and Taylor expansion.*

Once these notions are introduced, I will explain how it is possible to express this computational paradigm in a linear setting through a syntactical Taylor expansion. The idea is to understand exponential boxes in a differential variant of linear logic, and to represent it with linear combination.

If we have time, I will try to give an idea of some algebraic issues concerning this construction, and a method to show for example, that the normal form of the Taylor expansion of a MELL always converges.

NB: Taylor expansion is here analogical to the lambda calculus (with its differential version too) one, if someone heard about it, it can give a first intuition.

Séminaire des doctorants

Mercredi 15 novembre 2017, 11 heures, Salle 3052

**Chaitanya Leena Subramaniam** (“Algebra and calculus” and “proofs and programs” teams) *Homotopy type theory and the fibred structure of dependent types*

Chaitanya will do a 30 minutes talk. Given that the talk will not last as long as usual, we will also take advantage of the opportunity to discuss the organization of the seminar.

Abstract:

The talk will not be about my particular research problem. It will seek instead to give a gentle pictorial introduction to the inherent structure of dependent types and how this structure determines what dependent types and dependently-typed programs (or proofs) *mean*.

The focus will be on why this structure naturally leads to homotopy type theory and univalence. As a bonus, and if time permits, there will be some remarks on univalence and extensionality.

Séminaire des doctorants

Mercredi 8 novembre 2017, 11 heures, Salle 3052

**Francesco Antonio Genco** (Technische Universität Wien) *Typing Parallelism and Communication through Hypersequents*

Séminaire des doctorants

Mercredi 25 octobre 2017, 11 heures, Salle 3052

**Cédric Ho Thanh & Isaac Konan** (Algebra and calculus team / Combinatorics team) *New PhD student introduction session*

Isaac Konan, Bijective proof and generalization of Siladic's theorem

In a recent paper, Dousse introduced a refinement of Siladic̀’s theorem on partitions, by using the method of weighted words, where the different parts could take 2 colours. The proof of that refined theorem used some recursive equations with q-series. In this presentation, I will give the big lines of a bijective proof of the Dousse’s theorem, moreover which could be extended on a coloring with more than 2 colours.

Cédric Ho Thanh, Opétopes, Réécriture, et Koszulité

Ma thèse consiste en 3 mots que je vais tenter d'expliquer.

Séminaire des doctorants

Mercredi 11 octobre 2017, 11 heures, Salle 3052

**Hadrien Batmalle** (Équipe Preuves et Programmes) *From Cohen's Forcing to Classical Realisability: A New Approach*

Du forcing à la réalisabilité classique: une nouvelle approche

La réalisabilité classique permet d'interpréter des théories mathématiques classiques, comme la théorie des ensembles ZF, dans divers modèles de calculs (lambda-calcul avec continuations, domaines…). L'intérêt est double: côté informatique, il s'agit d'extraire des interprétations calculatoires de preuves classiques; côté mathématique, on obtient de nouveaux modèles de ces théories classiques (les deux aspects étant intimement liés). Une grande partie de la recherche en réalisabilité classique étudie la structure de ces modèles, qui apparaissent comme une généralisation du forcing de Cohen. Nous nous intéresserons ici à une nouvelle méthode pour exporter des propriétés des modèles de forcing aux modèles de réalisabilité, qui permet de construire des interprétations de deux théories contradictoires dans un même modèle de calcul, ce qui ouvre la voie à une analyse fine des conséquences calculatoires de la présence ou non de tel ou tel axiome.

From Cohen's Forcing to Classical Realisability: A New Approach

Classical realisability is a framework for interpreting classical theories in mathematics, such as the ZF set theory, in various models of computation (lambda-calculus with continuations, domains…). The goal is twofold: from the computer scientist's point of view, this is a method for extracting computational interpretations out of classical proofs; from the mathematician's, this is a trove of new models for these classical theories (both sides being tightly interwoven). A good deal of the research in this area is focussing on the structure of these models, arising as a generalisation of Cohen's forcing. In this talk we'll present some consequences of a new method for exporting properties of Cohen's forcing models into properties of classical realisability models. In particular we obtain interpretations of two incompatible theories in the same model of computation, which clears the path to studying the computational consequences of the presence, or lack thereof, of some axiom.

Séminaire des doctorants

Mercredi 27 septembre 2017, 11 heures, Salle 3052

**Baptiste Louf & Victor Lanvin** (Combinatorics and PPS teams) *New PhD student introduction session*

Combinatorial maps : algebraic and bijective enumeration

Combinatorial maps (which are embeddings of graphs on surfaces) are well studied objects in combinatorics, which have applications in other domains, such as quantum gravity. The goal is to enumerate them (sometimes exactly, sometimes asymptotically). For this purpose, one can resort to (among other things) bijective or algebraic methods. The algebraic method is often more powerful and yields results more easily, however bijections give a more in-depth understanding of the models. Often, formulas are found via powerful methods, then people try to re-prove them bijectively. In this talk, I present what I’m focusing on, on the bijective side (Carell-Chappy formula) and on the algebraic side (KP equations). If time permits, I will explain a simple bijection I discovered during my Master’s internship.

Gradual Set-Theoretic Types

A static type system can be an extremely powerful tool for a programmer, providing early error detection, and offering strong compile-time guarantees on the behavior of a program. However, compared to dynamic typing, static typing often comes at the expense of development speed and flexibility, as statically- typed code might be more difficult to adapt to changing requirements. Gradual typing is a recent and promising approach that tries to get the best of both worlds, by allowing the programmer to finely tune the distribution of dynamic and static checking over a program. However, this “gradualization” is sometimes too coarse, and an expression is often either fully dynamic or fully static. We argue that adding full-fledged union and intersection types (a.k.a. set-theoretic types) to a gradual type system solves this issue by making the transition between dynamic typing and static typing smoother.

Séminaire des doctorants

Mercredi 28 juin 2017, 11 heures, Salle 3052

**Tommaso Petrucciani** (PPS team) *Semantic subtyping: an introduction*

In the semantic subtyping approach, instead, types are interpreted as sets and subtyping is defined in terms of set containment. Then, an algorithm is derived from the definition. While the algorithm is complex, the interpretation of types serves as a fairly simple specification. This approach also ensures that union and intersection on types behave as the corresponding operations on sets.

I will give an introduction to this approach and show how to define subtyping semantically for types including arrows, union, intersection, and negation, following [Frisch et al., 2008]. Then, we will look at ongoing work on adapting this approach (originally studied for call-by-value languages) to lazy semantics.

[Frisch et al., 2008] A. Frisch, G. Castagna, and V. Benzaken, Semantic subtyping, JACM, 2008.

Séminaire des doctorants

Mercredi 14 juin 2017, 11 heures, Salle 3052

**Timo Zijlstra & Emmanuel Arrighi** *Quantum algorithms and Learning With Errors- based Cryptography & Distance Labels and Tree Skeletons*

Post quantum cryptography is meant to replace today's standards like RSA and Elliptic Curve Cryptography (ECC), since these standards are threathened by quantum algorithms. The most researched post-quantum candidates are based on lattice problems, and in particular the Learning with Errors (LWE) problem. It is assumed that there exists no quantum algorithm that solves this problem efficiently. However, in a particular setting and under some strong hypothesis, it is very easy to solve LWE using a generalization of the Bernstein-Vazirani quantum algorithm. We will take a look at possible quantum cryptanalysis on LWE-based cryptographic applications.

Distance Labels and Tree Skeletons:

To answer distance queries on a fix known graph, it is interesting to do precalculation in order to reduce query time. A methode is to use Hub Labeling. Hub Labeling works well on road transport network. We will take a look at this methode and introduce the notion of Skeleton Dimension which give an insight on why it works well on road network.

Séminaire des doctorants

Mercredi 31 mai 2017, 11 heures, Salle 3052

**Clément Jacq** (PPS team) *A playful introduction to game semantics (category-light)*

Une introduction ludique à la sémantique des jeux (allégée en catégories)

La sémantique des jeux est une branche de la théorie des modèles dont l'objectif est d'interpréter des formules de certaines logiques sous forme de jeux à deux joueurs. Son objectif initial était de lier les notions de vérité et de validité à des concepts de théorie des jeux tels que l'existence de stratégies gagnantes…

Après quelques exemples historiques, nous nous intéresserons dans cet exposé de manière informelle à un cas plus récent ou la sémantique des jeux modélise désormais des langages de programmation.

En guise de conclusion, nous évoquerons l'aspect formel avec la notion de structure catégorielle.

A playful introduction to game semantics (category-light)

Game semantics is a branch of model theory that aims at interpreting formulas of a given logic as two-player games. Initially, it was developed to link the notions of truth and validity to game-theoretic notions such as the existence of winning strategies.

After an historical example, we will look informally at a more recent case of game semantics where the games are used to model programming languages.

At the end, we'll mention the formal part with the notion of categorical model.

Séminaire des doctorants

Mercredi 17 mai 2017, 11 heures, Salle 3052

**Pierre Vial** (PPS team) *An Introduction to Intersection Type Systems, and a New Answer to Klop's Problem*

L'exposé aura deux buts:

1) Présenter les systèmes de types-intersection (ITS, intersection type systems), en particulier, les ITS à intersection non-idempotente. Je commencerai par des rappels basiques en lambda-calcul. On verra en quoi la représentation des lambda-termes par des arbres (bien qu'élémentaire) permet de comprendre la façon dont les ITS sont conçus et vérifient naturellement des propriétés que les systèmes de types simples ne peuvent (raisonnablement) pas avoir. Par exemple, dans un ITS, un terme est normalisable (i.e. il termine) ssi il est typable. Par opposition, dans un système de types simples, on aura seulement l'implication “Si le terme t est typable, alors il est normalisable” (*Propriété de Terminaison*). La notion de normalisation (i.e. terminaison) admet de nombreuses variantes: on en verra deux, la réduction de tête (HN, Head Normalization) et la réduction faible (WN, Weak Normalization). On verra aussi que les ITS ont des conséquences en lambda-calcul qui sont *externes* à la théorie des types.

Etant donné un système de type Sys (que Sys soit un ITS ou un système de types simples, d'ordre supérieur ou non), la propriété de terminaison (typable dans Sys ⇒ normalisable) est souvent difficile à établir (on doit généralement recourir à un argument dit de réalisabilité, attribué à Tait). Cependant, je présenterai le système R, introduit par Gardner et de de Carvalho, dans lequel l'opérateur d'intersection peut être vu comme non-idempotent et la terminaison repose sur un argument très simple que nous verrons ensemble.

2) Présenter un article (accepté récemment à LICS) traitant de lambda-calcul et de réduction infinitaires et dont voici l'abstract:

Infinitary Intersection Types as Sequences: a New Answer to Klop’s Problem

We provide a type-theoretical characterization of weakly-normalizing terms in an infinitary lambda-calculus. We adapt for this purpose the standard quantitative (with non-idempotent intersections) type assignment system of the lambda-calculus to our infinite calculus. Our work provides a positive answer to a semi-open question known as Klop’s Problem, namely, finding out if there is a type system characterizing the set of hereditary head-normalizing (HHN) lambda-terms. Tatsuta showed in 2007 that HHN could not be characterized by a finite type system. We prove that an infinitary type system endowed with a validity condition called approximability can achieve it. As it turns out, approximability cannot be expressed when intersection is represented by means of multisets. Multisets are then replaced coinductively by sequences of types indexed by integers, thus defining a type system called System S.

Séminaire des doctorants

Mercredi 3 mai 2017, 11 heures, Salle 3052

**Alexandre Nolin** (Algorithms and complexity Group) *Quantum, a look through nonlocality*

Séminaire des doctorants

Mercredi 19 avril 2017, 11 heures, Salle 3052

**Pierre Cagne** (PPS team) *Lawvere’s hyperdoctrines and notions of equality*

Séminaire des doctorants

Mercredi 5 avril 2017, 11 heures, Salle 3052

**Guillaume Lagarde** (Automata and applications Group) *On the stability of the Lempel-Ziv compression algorithm*

The presentation will just use basic combinatorics.

Séminaire des doctorants

Mercredi 22 mars 2017, 11 heures, Salle 3052

**Gabriel Radanne** (PPS team) *GADTs gone mild*

Since their adoption in “mainstream” languages, GADTs have been known for allowing to elegantly write toy typed interpreter at the cost of horrible type error messages and numerous headaches. Or, as Yaron Misky said, “I assumed that it was the kind of nonsense you get when you let compiler writers design your programming language.”.

In this talk, I will present GADTs, what they are, and what useful things we can do with them. This will take us on quite a journey, with some traces of C, a pinch of memory layout, a cameo from pushdown automata and a healthy amount of Prolog. The only requirements will be a passing familiarity with OCaml and the caffeinated beverage of your choice.

Séminaire des doctorants

Mercredi 8 mars 2017, 11 heures, Salle 3052

**Pablo Eduardo Rotondo** (Automata and applications Group) *Continued Fractions and the Recurrence of Sturmian Words*

Séminaire des doctorants

Mercredi 22 février 2017, 11 heures, Salle 3052

**Lucas Boczkowski** (Algorithms and complexity Group) *Minimizing Message Size in Stochastic Communication Patterns: Fast Self-Stabilizing Protocols with 3 bits*

This paper considers the basic PULL model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of n agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast (also termed Rumor Spreading). We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. In particular, the specification of who is a source and what is its initial input bit may be set by an adversary.

We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property (called “the bitwise-independence property”) that uses l-bits messages to one that uses only (log l)-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer T, within O(log n log T) rounds w.h.p., and using messages that contain 3 bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing majority broadcast protocol which converges in O(log n) time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.

Based on joint work with A. Korman and E. Natale.

Séminaire des doctorants

Mercredi 25 janvier 2017, 11 heures, Salle 3052

**Thibaut Girka** (PPS team) *Oracle-based Differential Operational Semantics (or Explaining program differences with programs)*

We illustrate this framework by instantiating it on a toy imperative language and by presenting several /difference languages/ ranging from trivial equivalence-preserving syntactic transformations to characterized semantic differences. Through those examples, we will present the basis of our framework, show how to use it to relate syntactic changes with their effect on semantics, how one can abstract away from the small-step semantics presentation, and discuss the composability of oracles.

Séminaire des doctorants

Mercredi 11 janvier 2017, 11 heures, Salle 3052

**Fabian Reiter** (Automata and applications Group) *Asynchronous Distributed Automata*