It is widely known that the precision of a program analyzer is closely related to intensional program properties, namely, properties concerning how the program is written. This explains, for instance, the interest in code obfuscation techniques, namely, tools explicitly designed to degrade the results of program analysis by operating syntactic program transformations. Less is known about a possible relation between what the program extensionally computes, namely, its input-output relation, and the precision of a program analyzer. In this paper we explore this potential connection in an effort to isolate program fragments that can be precisely analyzed by abstract interpretation, namely, programs for which there exists a complete abstract interpretation. In the field of static inference of numeric invariants, this happens for programs, or parts of programs, that manifest a monotone (either non-decreasing or non-increasing) behavior. We first formalize the notion of program monotonicity with respect to a given input and a set of numerical variables of interest. A sound proof system is then introduced with judgments specifying whether a program is monotone relatively to a set of variables and a set of inputs. The interest in monotonicity is justified because we prove that the family of monotone programs admit a complete abstract interpretation over a specific class of non-trivial numerical abstractions and inputs. This class includes all non-relational abstract domains that refine interval analysis (i.e., at least as precise as the intervals abstraction) and that satisfy a topological convexity hypothesis.

Jury :
- Claire Mathieu, présidente, directrice de recherche au CNRS, Académie des sciences
- Meena Mahajan, rapporteuse, professeure à l'Institut des sciences mathématiques de Chennai
- Santiago Figueira, rapporteur, professeur à l'université de Buenos Aires
- Olivier Bournez, rapporteur, professeur à l'École polytechnique
- Damiano Mazza, examinateur, directeur de recherche au CNRS
- Olivier Carton, examinateur, professeur à l'université Paris Cité, IUF

« Résumé opérationnel » :
Dans une première partie, nous comparerons quelques méthodes efficaces de compression (automates à pile, LZ'78, espace polylogarithmique), puis nous étudierons la « catastrophe du premier bit » pour l'algorithme de Lempel-Ziv, où l'ajout d'un seul bit peut changer le taux de compression. La seconde partie traitera de complexité algébrique, en particulier de bornes inférieures pour certains modèles de calcul de polynômes, et du célèbre problème de test d'identité polynomiale qui résiste encore aux tentatives de dérandomisation.

In the LOCAL model of distributed computing, n processors are connected together through a graph. They communicate with their neighbours in a synchronised way a limited number of times (through infinite bandwith communication channels), and then produce an output. Their goal is to find a strategy solving some information processing task, e.g. to obtain a proper coloring of the graph. To answer this question, the considered physical theory of information matters: for instance, Quantum Information Theory (QIT) can in principle allow for strategies solving the task quicker.

Finding lower bounds on the maximal speed at which QIT can solve a task is tricky. To circumvent this difficulty, several proofs based on the No-Signalling (NS) Principle (which states that an event A can influence an event B only if some signal can physically be sent from A to B) were proposed and a new NS-LOCAL model of distributed computing was proposed.

I’ll re-interpret this NS-LOCAL model based on the theoretical physics formalism of light cones in circuits. I will justify that the NS-LOCAL model is the “largest reasonable model with a pre-shared resource between the nodes”. Then, I’ll introduce a weaker Causal-LOCAL model, the “largest reasonable model without pre-shared resource between the nodes”.

All this will be illustrated with easy to understand examples. I was recently told by Vaclav Rozhon: We computer scientists are very scared of the word “quantum”. Note that I’ll almost not talk about quantum theory, hence no need to be scared.

Conférence à l'IHP incluant une journée spéciale “Séminaire Flajolet” le jeudi 7.

Inscription obligatoire sur le site de la conférence: https://indico.math.cnrs.fr/event/8115/registrations/

Voir aussi: https://semflajolet.math.cnrs.fr/

In this presentation, I will provide an introduction to proof theory within the context of non-wellfounded proof systems. A non-wellfounded proof is generated by finitely branching logical rules, but potentially resulting in an infinitely deep proof tree. After defining the system, I will present a proof of cut-elimination for the $\mu$-MALL system, drawing from the work of Baelde et al. in 2016. Furthermore, I will discuss how we aim to extend this result with Alexis Saurin.

Primary key constraints in databases state there can be at most one tuple for every primary key in a given relation. However, these days it is common to have situations where this property cannot be maintained. Databases that violate constraints are called « inconsistent databases ». In particular, if a database violates primary key constraint, it will contain more than one tuple for the same primary key. In this setting, the notion of a repair is defined by picking exactly one tuple for each primary key (maximal consistent subset of the database). A Boolean conjunctive query q is certain for an inconsistent database D if q evaluates to true over all repairs of D. In this context, there is a dichotomy conjecture that states that for a fixed boolean conjunctive query q, testing whether q is certain for an input database D is either polynomial time or coNP-complete.

The conjecture is open in general and has been open for more than two decades. In this talk we will introduce two new algorithms that we have devised, one based on fix-point computation and the other based on bipartite matching. We will also see how these algorithms can be used to prove the dichotomy for the cases that were previously open.

These results were obtained in collaboration with Diego Figueira, Luc Segoufin and Cristina Sirangelo and combine the results that were presented at ICDT 2023 and what will be presented at PODS 2024.

the dichromatic number of a digraph is the minimum integer k such that the set of vertices of D can be partitioned into k acyclic subdigraphs. It is easy to see that the chromatic number of a graph G is the same as the dichromatic of the digraph obtained from G by replacing each edge by a digon (two anti-parallel arcs). Based on this simple observation, many theorems concerned with chromatic number of undirected graphs have been generalised to digraphs via the dichromatic number. However, no concept of clique number for digraphs was available. The purpose of this presentation is to explore such a concept and its relationship with the dichromatic number, mirroring the relationship between the clique number and chromatic number in undirected graphs. We will focus on studying the notion of chi-boundedness in particular. This is a joint work with Guillaume Aubian, Pierre Charbit and Raul Lopes.