The aim of the CARTE research team is to take into account adversity in computations, which is implied by actors whose behaviors are unknown or unclear. We call this notion adversary computation.

The project combines two approaches. The first one is the analysis of the behavior of systems, using tools coming from Continuous Computation Theory. The second approach is to build defenses with tools coming from logic, rewriting and, more generally, from Programming Theory.

The activities of the CARTE team are organized around two research actions:

Computation over Continuous Structures

Computer Virology.

From a historical point of view, the first official virus appeared in 1983 on Vax-PDP 11. At the same time, a series of papers was published which always remains a reference in computer virology: Thompson , Cohen and Adleman . The literature which explains and discusses practical issues is quite extensive , . However, there are only a few theoretical/scientific studies, which attempt to give a model of computer viruses.

A virus is essentially a self-replicating program inside an
adversary environment. Self-replication has a solid background
based on works on fixed point in

a virus infects programs by modifying them,

a virus copies itself and can mutate,

it spreads throughout a system.

The above scientific foundation justifies our position to use the word virus as a generic word for self-replicating malwares. There is yet a difference. A malware has a payload, and virus may not have one. For example, a worm is an autonomous self-replicating malware and so falls into our definition. In fact, the current malware taxonomy (virus, worms, trojans, ...) is unclear and subject to debate.

Classical recursion theory deals with computability over discrete structures (natural numbers, finite symbolic words). There is a growing community of researchers working on the extension of this theory to continuous structures arising in mathematics. One goal is to give foundations of numerical analysis, by studying the limitations of machines in terms of computability or complexity, when computing with real numbers. Classical questions are : if a function

While the notion of a computable function over discrete data is captured by the model of Turing machines, the situation is more delicate when the data are continuous, and several non-equivalent models exist. In this case, let us mention computable analysis, which relates computability to topology , ; the Blum-Shub-Smale model (BSS), where the real numbers are treated as elementary entities ; the General Purpose Analog Computer (GPAC) introduced by Shannon with continuous time.

The rewriting paradigm is now widely used for specifying, modelizing, programming and proving. It allows one to easily express deduction systems in a declarative way, and to express complex relations on infinite sets of states in a finite way, provided they are countable. Programming languages and environments with a rewriting based semantics have been developed ; see ASF+SDF , Maude , and Tom .

For basic rewriting, many techniques have been developed to prove properties of rewrite systems like confluence, completeness, consistency or various notions of termination. Proof methods have also been proposed for extensions of rewriting such as equational extensions, consisting of rewriting modulo a set of axioms, conditional extensions where rules are applied under certain conditions only, typed extensions, where rules are applied only if there is a type correspondence between the rule and the term to be rewritten, and constrained extensions, where rules are enriched by formulas to be satisfied , , .

An interesting aspect of the rewriting paradigm is that it allows automatable or semi-automatable correctness proofs for systems or programs: the properties of rewriting systems as those cited above are translatable to the deduction systems or programs they formalize and the proof techniques may directly apply to them.

Another interesting aspect is that it allows characteristics or properties of the modeled systems to be expressed as equational theorems, often automatically provable using the rewriting mechanism itself or induction techniques based on completion . Note that the rewriting and the completion mechanisms also enable transformation and simplification of formal systems or programs.

Applications of rewriting-based proofs to computer security are various. Approaches using rule-based specifications have recently been proposed for detection of computer viruses , . For several years, in our team, we have also been working in this direction. We already proposed an approach using rewriting techniques to abstract program behaviors for detecting suspicious or malicious programs , .

It is rightful to wonder why there are only a few fundamental studies on computer viruses while it is one of the important flaws in software engineering. The lack of theoretical studies explains maybe the weakness in the anticipation of computer diseases and the difficulty to improve defenses. For these reasons, we do think that it is worth exploring fundamental aspects, and in particular self-reproducing behaviors.

The crucial question is how to detect viruses or self-replicating malwares. Cohen demonstrated that this question is undecidable. The anti-virus heuristics are based on two methods. The first one consists in searching for virus signatures. A signature is a regular expression, which identifies a family of viruses. There are obvious defects. For example, an unknown virus will not be detected, like ones related to a 0-day exploit. We strongly suggest to have a look at the independent audit in order to understand the limits of this method. The second one consists in analyzing the behavior of a program by monitoring it. Following , this kind of methods is not yet really implemented. Moreover, the large number of false-positive implies this is barely usable. To end this short survey, intrusion detection encompasses virus detection. However, unlike computer virology, which has a solid scientific foundation as we have seen, the IDS notion of “malwares” with respect to some security policy is not well defined. The interested reader may consult .

The aim is to define security policies in order to prevent malware propagation. For this, we need (i) to define what is a computer in different programming languages and setting, (ii) to take into consideration resources like time and space. We think that formal methods like rewriting, type theory, logic, or formal languages, should help to define the notion of a formal immune system, which defines a certified protection.

This study on computer virology leads us to propose and construct a “high security lab” in which experiments can be done in respect with the French law.

Understanding computation theories for continuous systems leads to studying hardness of verification and
control of these systems. This has been used to discuss problems in fields as diverse as verification (see e.g.,
), control theory (see e.g., ), neural
networks (see e.g., ), and so on.
We are interested in the formal decidability of properties of
dynamical systems, such as reachability ,
the Skolem-Pisot problem , the computability of
the

Contrary to computability theory, complexity theory over continuous spaces is underdeveloped and not well understood. A central issue is the choice of the representation of objects by discrete data and its effects on the induced complexity notions. As for computability, it is well known that a representation is gauged by the topology it induces. However more structure is needed to capture the complexity notions: topologically equivalent representations may induce different classes of polynomial-time computable objects, e.g., developing a sound complexity theory over continuous structures would enable us to make abstract computability results more applicable by analyzing the corresponding complexity issues. We think that the preliminary step towards such a theory is the development of higher-order complexity, which we are currently carrying out.

In contrast with the discrete setting, it is of utmost importance to compare the various models of computation over the reals, as well as their associated complexity theories. In particular, we focus on the General Purpose Analog Computer of Claude Shannon , on recursive analysis , on the algebraic approach and on Markov computability . A crucial point for future investigations is to fill the gap between continuous and discrete computational models. This is one deep motivation of our work on computation theories for continuous systems.

The other research direction on dynamical systems we are interested in is the study of properties of adversary systems or programs, i.e., of systems whose behavior is unknown or indistinct, or which do not have classical expected properties. We would like to offer proof and verification tools, to guarantee the correctness of such systems. On one hand, we are interested in continuous and hybrid systems. In a mathematical sense, a hybrid system can be seen as a dynamical system, whose transition function does not satisfy the classical regularity hypotheses, like continuity, or continuity of its derivative. The properties to be verified are often expressed as reachability properties. For example, a safety property is often equivalent to (non-)reachability of a subset of unsure states from an initial configuration, or to stability (with its numerous variants like asymptotic stability, local stability, mortality, etc ...). Thus we will essentially focus on verification of these properties in various classes of dynamical systems.

We are also interested in rewriting techniques, used to describe dynamic systems, in particular in the adversary context. As they were initially developed in the context of automated deduction, the rewriting proof techniques, although now numerous, are not yet adapted to the complex framework of modelization and programming. An important stake in the domain is then to enrich them to provide realistic validation tools, both in providing finer rewriting formalisms and their associated proof techniques, and in developing new validation concepts in the adversary case, i.e., when usual properties of the systems like, for example, termination are not verified. For several years, we have been developing specific procedures for property proofs of rewriting, for the sake of programming, in particular with an inductive technique, already applied with success to termination under strategies , , , to weak termination , sufficient completeness and probabilistic termination . The last three results take place in the context of adversary computations, since they allow for proving that even a divergent program, in the sense where it does not terminate, can give the expected results. A common mechanism has been extracted from the above works, providing a generic inductive proof framework for properties of reduction relations, which can be parametrized by the property to be proved , . Provided program code can be translated into rule-based specifications, this approach can be applied to correctness proof of software in a larger context. A crucial element of safety and security of software systems is the problem of resources. We are working in the field of Implicit Computational Complexity. Interpretation based methods like Quasi-interpretations (QI) or sup-interpretations, are the approach we have been developing these last years , , . Implicit complexity is an approach to the analysis of the resources that are used by a program. Its tools come essentially from proof theory. The aim is to compile a program while certifying its complexity.

Hugo Férée has received the Ackermann award for his PhD thesis “complexité d'ordre supérieur et analyse récursive”.

Functional Description

Codisasm is a new disassembly program which supports self-modifying code and code overlapping. Up to our knowledge, it is the first which copes both aspects of program obfuscation. The tool is based on the notion of “wave” developed in the group.

It is written in C and contains about 3k lines of code.

Contact: Fabrice Sabatier

Functional Description

DynamicTracer is a new tool with a public web interface which provides run traces of executable files. The trace is obtained by recording a dynamic execution in a safe environment. It contains instruction addresses, instruction opcodes and other optional information.

It is written in C++ and contains about 2.5k lines of code.

Contact: Fabrice Sabatier

Functional Description

Gorille (formerly MMDEX) is a virus detector based on morphological analysis. It is composed of our own disassembler tool, of a graph transformer and a specific tree-automaton implementation. The tool is used in the EU-Fiware project and by some other partners (e.g., DAVFI project).

It is written in C and contains about 100k lines of code.

APP License, IDDN.FR.001.300033.000.R.P.2009.000.10000, 2009.

Contact: Philippe Antoine

**Complexity of stream functions and higher-order complexity.**
We have pursued our works on higher-order complexity and the complexity of stream functions. Both notions are closely related as any function from natural numbers to natural numbers can be seen as a stream (an infinite list) of natural numbers:

A characterization of the class of Basic Feasible Functionals using term rewrite systems on streams and interpretation methods has been proposed in . This result is part of Hugo Férée's PhD thesis for which he has obtained the Ackermann award.

**Complexity analysis of Object-Oriented programs.**
We have proposed a type system based on non-interference and data ramification (tiering) principles in . We have captured the set of functions computable in polynomial time on OO programs. The studied language is general enough to capture most OO constructs and the characterization
is quite expressive as it allows the analysis of a combination of imperative loops and of data ramification scheme based on Bellantoni and Cook’s safe recursion using function algebra.

**Rice-like theorem for primitive recursive functions.**
We have studied the following question: what are the properties of primitive recursive functions that are decidable (by a Turing machine), given a primitive recursive presentation of the function. We give a complete characterization of these properties. We show that they can be expressed as unions of elementary properties of being compressible. If

**Parametrization of geometric figures.**
During the master internship of Diego Nava Saucedo, we have studied the semi-computability of geometric figures. A figure is semi-computable if there is a program that semi-decides whether a pixel intersects the figure. Our goal is to understand the semi-computability of a figure in terms of the parameters describing the figure. It turns out that the usual ways of parameterizing simple figures such as triangles, squares or disks do not behave well in terms of semi-computability. We have actually proved that no *finite* parametrization behaves well.

**Symbolic Dynamics on Groups.**
In an effort to better understand the interplay of geometry and
computability in tiling theory, E. Jeandel has studied tiling problems
on general Cayley graphs, and has obtained a significant number of new
results.
He has proven that groups with an (strongly) aperiodic tiling system
have decidable word problem , and provided
examples of new groups (in particular monster groups) with such tiling
systems, and proved that all nontrivial nilpotent groups have an
aperiodic tiling system and an undecidable domino problem
.
He also showed how the new concept of translation-like actions from
geometric group theory can be used to prove that many groups, in
particular the Grigorchuk groups and most groups with a nontrivial center,
have an undecidable domino problem .

**The smallest aperiodic tileset.**
In a joint work with Michael Rao, E. Jeandel has proven that there
exists an aperiocic set of 11 Wang tiles , and furthermore
that this number is optimal.

**The ZX Calculus is incomplete for Clifford+T quantum mechanics.**
The ZX calculus is a diagrammatic language for quantum mechanics and quantum information processing. In this paper, Simon Perdrix and Harny Wang prove that the ZX-calculus is not complete for the Clifford+T quantum mechanics. The completeness for this fragment has been stated as one of the main current open problems in categorical quantum mechanics. The ZX calculus was known to be incomplete for quantum mechanics, on the other hand, it has been proved complete for Clifford quantum mechanics (a.k.a. stabilizer quantum mechanics), and for single-qubit Clifford+T quantum mechanics. The question of the completeness of the ZX calculus for Clifford+T is a crucial step in the development of the ZX calculus because of its (approximate) universality for quantum mechanics (i.e. any unitary evolution can be approximated using Clifford and T gates only). They exhibit a property which is know to be true in Clifford+T quantum mechanics and prove that this equation cannot be derived in the ZX calculus, by introducing a new sound interpretation of the ZX calculus in which this particular property does not hold. Finally, we propose to extend the language with a new axiom.
This result has been presented as invited speakers in the conferences "Quantum Theory: from foundations to technologies" in Vaxjo Sweden, and "Higher TQFT and categorical quantum mechanics” at the Scrounger Institute in Vienna. The authors also presented these results at the workshop of the CNRS groupe de travail Informatique Quantique du GDR IM, in Grenoble.

**Block Representation of Reversible Causal Graph Dynamics.**
In this work presented at the conference on Foundation of computer science (FCT’15) , Pablo Arrighi, Simon Martiel and Simon Perdrix, consider a reversible version of the causal graph dynamics. Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a further physics-like symmetry, namely reversibility. More precisely, we show that Reversible Causal Graph Dynamics can be represented as finite-depth circuits of local reversible gates.

**Reversibility in the Extended Measurement-based Quantum Computation.**
In this work by Nidal Hamrit and Simon Perdrix has been presented at the conference on Reversible Computation in Grenoble . When applied on some particular quantum entangled states, measurements are universal for quantum computing. In particular, despite the fondamental probabilistic evolution of quantum measurements, any unitary evolution can be simulated by a measurement-based quantum computer (MBQC). They consider the extended version of the MBQC where each measurement can occur not only in the X,Y-plane of the Bloch sphere but also in the X,Z- and Y,Z-planes. The existence of a gflow in the underlying graph of the computation is a necessary and sufficient condition for a certain kind of determinism. They extend the focused gflow (a gflow in a particular normal form) defined for the X,Y-plane to the extended case, and provide necessary and sufficient conditions for the existence of such normal forms.

**Quantum Circuits for the Unitary Permutation Problem.**
In this paper presented at TAMC’15 Stefano Facchni and Simon Perdrix consider the *Unitary Permutation* problem which consists, given *permutation as substring* problem introduced by Karp.

Simon Perdrix is the principal investigator of the project “measurement-based quantum computing” funded by Région Lorraine and Université de Lorraine.

The team is a funding partner in ANR Elica (2014-2019), "Elargir les idées logistiques pour l'analyse de complexité". The Carte team is well-known for its expertise in implicit computational complexity.

The team is a funding partner in ANR Binsec (2013-2017), whose aim is to fill part of the gap between formal methods over executable code, and binary-level security analyses currently used in the security industry. Two main applicative domains are targeted: vulnerability analysis and virus detection. Two other closely related applications will also be investigated: crash analysis and program deobfuscation.

Submission of an Inria associate team proposal ACRA (Applications of Complexity to Resource Analysis) in collaboration with Computer Science and Engineering department, State University New York, Buffalo. The french principal investigator is Romain Péchoux, the US principal investigator is Marco Gaboardi.

An Hubert Curien Partnership (PHC) PHC Imhotep from the French Ministry of Foreign Affairs and with the support of the French Ministry of National Education and Ministry of Higher Education and Research holds between members of EPC Carte and Alexandria E-Just University.

Foundations of Quantum Computation: Syntax and Semantics (FoQCoSS), Regional Program STIC-AmSud. This 2-year project has been accepted in late 2015. The Argentinian-Brazilian-French consortium consists of: Pablo ARRIGHI (Université Aix-Marseille, France), Alejandro DIAZ-CARO (Universidad Nacional de Quilmes, Argentina), Gilles DOWEK (Inria, France), Juliana KAIZER VIZZOTTO (Universidade Federal de Santa Maria, Brazil), Simon PERDRIX (CNRS/Carte, France) and Benoît VALIRON (CentraleSupélec – LRI, France). The ultimate goal of this project is to study the foundations of quantum programming languages and related formalisms. With this goal in mind, we will study topics such as parallelism, probabilistic systems, isomorphisms, etc. The interest goes beyond having a working programming language for quantum computing; we are interested, on one hand, in its individual characteristics and its consequences for classical systems, and, on the other hand, in its implications for the foundations of quantum physics.

Walid Gomaa, associate professor at Alexandria E-Just University, was invited during two months (April and November) in the team.

Daniel Leivant, professor at Indiana University in Bloomington, was invited in June and July.

Mizuhito Ogawa was invited in the group to discuss about models of self-modifying code based on pushdown automata. He came back in October for further collaboration.

Guillaume Bonfante and Jean-Yves Marion participated to the organisation of the partnership meeting between JAIST and LORIA in Nancy, October 2015

Simon Perdrix has been member of the organizing committee of the Workshop of the Groupe de Travail informatique Quantique in Grenoble 26-27 November 2015.

Simon Perdrix has been member of the organizing committee of the Workshop of the Groupe de Travail GeoCal-LAC-LTP in Nancy 12-14 October 2015.

Guillaume Bonfante was co-chair of the 8th international symposium on foundation and practice of security 2015.

Guillaume Bonfante was in the Program Committee of Protection and Reverse Engineering Workshop 2015.

Guillaume Bonfante was in the Program Committee of the workshop on Logic and Computational Complexity (LCC) 2015.

Emmanuel Jeandel was in the Program Committee of Computability in Europe (CiE) 2015.

Jean-Yves Marion was in the Program Committee of the 8th international symposium on foundation and practice of security 2015.

Jean-Yves Marion was in the Program Committee of Protection and Reverse Engineering Workshop 2015.

Romain Péchoux was in the Program Committee of FOundational and Practical Aspects of Resource Analysis (FOPARA) 2015.

Simon Perdrix was in the Program Committee of Asian Quantum Information Science Conference (AQIS) 2015.

Simon Perdrix is in the Program Committee of Quantum Physics and Logic (QPL), forthcoming: QPL’16 in Glasgow in June 2016.

Mathieu Hoyrup reviewed articles for:

MFCS 2015

STACS 2015

Emmanuel Jeandel reviewed articles for:

MFCS 2015

STACS 2015

Romain Péchoux reviewed articles for:

FOPARA 2015

ISMVL 2015

FOSSACS 2016

Simon Perdrix reviewed articles for:

AQIS 2015

ICALP 2015

LICS 2015

QPL 2016

Emmanuel Hainry reviewed articles for:

*Theoretical Computer Science*

*Applicable Analysis and Discrete Mathematics*

Mathieu Hoyrup reviewed articles for:

*Journal of Symbolic Logic*

*Information and Computation*

*Mathematical Structures in Computer Sciences*

*Logical Methods in Computer Science*

*Theory of Computing Systems*

Emmanuel Jeandel reviewed articles for:

*Discrete Mathematics and Theoretical Computer Science*

* Ergodic Theory and Dynamical Systems*

Romain Péchoux reviewed articles for:

*Computability - Journal of the association Computability In Europe*

*Information and Computation*

Simon Perdrix reviewed articles for:

*Quantum Information and Computation*

Guillaume Bonfante was invited to give a talk on implicit complexity within NC at the Shonan Meeting on Low Level Code Analysis and Application to Computer Security.

Mathieu Hoyrup was invited to give a talk at the annual workshop Continuity, Computability, Constructivity (CCC 2015) in Kochel am See, Germany, September 2015.

Simon Perdrix was invited to give a talk:

“The ZX Calculus is incomplete for Clifford+T quantum mechanics”, at "Quantum Theory: from foundations to technologies – QTFT", Vaxjo, Sweden, June 2015,

“Supplementary of Interacting Frobenius Algebras” at the Workshop on “Higher topological quantum field theory and categorical quantum mechanics” Erwin Schrödinger International Institute, Vienna, October 2015

“Informatique quantique et théorie des graphes” at Journées Graphes et Algorithmes 2015, Orléans, Novembre 2015.

Romain Péchoux is external expert for the European commission Horizon 2020 program.

Isabelle Gnaedig is:

vice-leader of the team Carte.

Emmanuel Hainry is:

member of the CNU (Conseil National des Universités), Section 27.

organizer of the Carte Seminar.

Mathieu Hoyrup is:

principal investigator of a PHC Imhotep with Walid Gomaa (Alexandria E-Just University).

organizer of the Formal Methods Seminar at Loria.

Romain Péchoux is:

responsible of the Project-team Carte activity report 2015.

principal investigator of the Inria associate team Acra proposal.

Simon Perdrix is:

responsible of GT IQ (groupe de travail Informatique quantique) at the CNRS GdR IM (groupe de recherche Informatique Mathématique).

Unless explicitly stated, the teachings below are given at Université de Lorraine.

Licence:

Guillaume Bonfante

Java, L3, Mines Nancy

Emmanuel Hainry

Operating Systems, 30h, L1, IUT Nancy Brabois

Algorithmics, 40h, L1, IUT Nancy Brabois

Dynamic Web, 60h, L1, IUT Nancy Brabois

Databases, 30h, L1, IUT Nancy Brabois

Object Oriented Languages, 12h, L2, IUT Nancy Brabois

Complexity, 30h, L2, IUT Nancy Brabois

Mathieu Hoyrup

Programmation C, 15h, L1 PACES

Programmation JAVA, 56h, L1, IUT Charlemagne

Emmanuel Jeandel

Algorithmics and Programming 1, 60h, L1 Maths-Info

Algorithmics and Programming 4, 30h, L3 Informatique

Modelling Using Graph Theory, 30h, L3 Informatique

Networking, 15h, L3 Informatique

Data Compression, 45h, L2 Informatique

Romain Péchoux

Programmation orientée objet, 61,5h, L3 MIASHS

Programmation orientée objet, 53,5h, L2 MIASHS

Outils logiques pour l'informatique, 35h, L1 MIASHS

Bases de données, 40h, L3 Sciences de la Gestion

Algorithmic complexity, 30h, L3 MIAGE, IGA Casablanca, Morocco.

Simon Perdrix

Structure de données, 72h, L1, IUT Charlemagne

Licence : Modelling Using Graph Theory, 15h, L3 Informatique

Master:

Guillaume Bonfante

Modelling and UML, M1, Mines Nancy

Video Games, M1, Mines Nancy

Semantics, M1, Mines Nancy

Safety of Software, M2, Mines Nancy

Isabelle Gnaedig

Design of Safe Software, Coordination of the module, M2, Telecom-Nancy

Rule-based Programming, 20h, M2, Telecom-Nancy

Emmanuel Hainry

Implicit Complexity, 15h, M2 Informatique

Emmanuel Jeandel

Algorithmics and Complexity, 30h, M1 Informatique

Combinatorial Optimization, 36h, M1 Informatique

Romain Péchoux

Mathematics for computer science, 30h, M1 SCA

Advanced Java, 52,5h, M1 MIAGE

Implicit Complexity, 15h, M2 Informatique

PhD in progress: Hubert Godfroy, Semantics of Self-modifying Programs, Jean-Yves Marion (director).

PhD in progress: David Cattanéo, Combinatorial Modelization in Quantum Computation and Generalized Cover Problems, Pablo Arrighi (director), Simon Perdrix (co-advisor).

PhD: Thanh Dinh Ta, 11 May 2015, Malware Algebraic Modeling and Detection, started Sept. 2010, Jean-Yves Marion (director) and Guillaume Bonfante (co-advisor).

PhD: Aurélien Thierry, 11 March 2015, Morphological Analysis of Malware, Jean-Yves Marion (director).

PhD in progress: Paul Bakouche, Mesure de complexité en topologie de petites dimensions, Florian Deloup (co-advisor) and Guillaume Bonfante (director).

Isabelle Gnaedig was:

member of the Inria hiring committee for young researchers,

member of the Telecom-Nancy engineering school admission committee.

Emmanuel Jeandel was:

Reviewer and Examiner of Simon Martiel's PhD Defense on "Approches informatique et mathématique des dynamiques causales de graphes", defended in Université de Nice Sophia Antipolis, July 6th 2015

Reviewer of Ilkka Törmä's PhD on "Structural and Computational Existence Results for Multidimensional Subshifts", defended in Turku (Finland), July 31st 2015

Reviewer of Rodrigo Torres' PhD on "Some Dynamical Properties of Turing Machines Dynamical Models", to be defended in January 2016, Santiago de Chile (Chile)

Isabelle Gnaedig is member of the scientific vulgarization committee of Inria Nancy - Grand Est. This committee is a choice and guidance instance helping the direction of the center and the person in charge of popularization events to elaborate a strategy, to realize events and to help researchers to get involved in various actions aiming at popularizing our research themes, and more generally computer science and mathematics.

Mathieu Hoyrup wrote an article on his recent work for Images des Mathématiques.