The talk will start with presenting a brief overview of querying incomplete information in databases and the main computational challenges it presents. Querying incomplete data invariably relies on the very coarse classification of query answers into those that are certain and those that are not. Such a classification is often very costly, and we propose to refine it by measuring how close an answer is to certainty.

This measure is defined as the probability that the query is true under a random interpretation of missing information in a database. Since there are infinitely many such interpretations, to pick one at random we adopt the approach used in the study of asymptotic properties and 0-1 laws for logical sentences and define the measure as the limit of a sequence. We show that in the standard model of missing data, the 0-1 law is observed: this limit always exists and can be only 0 or 1 for a very large class of queries. Thus, query answers are either almost certainly true, or almost certainly false, and this classification behaves very well computationally. When databases satisfy constraints, the measure is defined as the conditional probability of the query being true if the constraints are true. This can now be an arbitrary rational number, which is always computable. Another refinement of the notion of certainty views answers with a larger set of interpretations that make them true as better ones. We pinpoint the exact complexity of finding best answers for first-order queries.

En routage next-hop, paradigme de routage utilisé dans l'Internet Global, chaque routeur choisit le next-hop de chaque paquet en fonction de son adresse destination. Le routage sensible à la source est une extension compatible du routage next-hop où le choix du next-hop dépend de l'adresse source du paquet en plus de son adresse destination. Nous montrons dans cette thèse que le routage sensible à la source est adapté au routage des réseaux multihomés avec plusieurs adresses, qu'il est possible d'étendre de manière compatible les protocoles de routage à vecteur de distance existants et que ce paradigme de routage offre avantageusement plus de flexibilité aux hôtes. Nous montrons d'abord que certains systèmes n'ordonnent pas correctement les entrées sensibles à la source dans leurs tables de routage et nous définissons un algorithme adapté aux protocoles de routage pour y remédier. Nous montrons comment étendre les protocoles à vecteur de distances au routage sensible à la source de manière compatible. Nous validons notre approche en concevant une extension d'un protocole existant (Babel), en réalisant la première implémentation complète d'un protocole sensible à la source et en utilisant ce protocole pour router un réseau multihomé. Enfin, nous montrons que le routage sensible à la source offre des possibilités de multichemin aux couches supérieures des hôtes. Nous vérifions qu'il s'intègre aux technologies existantes (MPTCP) et nous concevons des techniques d'optimisation pour les applications légères. Nous évaluons ces techniques après les avoir implémentées dans le cadre d'une application existante (mosh).

One great issue in conceiving and specifying distributed algorithms is the sheer number of models, differing in subtle and often qualitative ways: synchrony, kind of faults, number of faults… In the context of message-passing, one solution is to restrict communication to proceed by round; A great variety of models can then be captured in the Heard-Of model, with predicates on the communication graph at each round. However, this raises another issue: how to characterize a given model by such a predicate? It depends on how to implement rounds in the model. This is straightforward in synchronous models, thanks to the upper bound on communication delay. On the other hand, asynchronous models allow unbounded message delays, which makes the implementation of rounds dependent on the specific message-passing model.

I will present our formalization of this characterization for asynchronous models. Specifically, we introduce Delivered collections: the collection of all messages delivered at each round, whether late or not. Defining predicates on Delivered collections then allows us to capture message-passing models at the same level of abstraction than Heard-Of predicates. The question is then reformulated to: what Heard-Of predicates can be generated by a given Delivered predicate?

I will provide an answer by considering all possible scheduling of deliveries of messages from the Delivered collections and change of rounds for the processes. Strategies of processes then constrain those scheduling by specifying when processes can change rounds; those ensuring no process is ever blocked forever generate a Heard-Of collection per run, that is a Heard-Of predicate. Finally, we use these strategies to nd a characterizing Heard-Of predicate through a dominance relation on strategies: a dominant strategy for a Delivered predicate implements the most constrained Heard-Of predicate possible. This approach oer both the dominant Heard-Of predicates for classical asynchronous models and the existence, for every Delivered predicate, of a strategy dominating large classes of strategies. On the whole, those results confirm the power of this formalization and demonstrate the characterization of asynchronous models through rounds as a worthwhile pursuit.

This is joint work with Aurélie Hurault and Philippe Quéinnec.

The topic of this thesis is the study of the encoding of references and concurrency in Linear Logic. Our perspective is to demonstrate the capability of Linear Logic to encode side-effects to make it a viable, formalized and well studied compilation target for functional languages in the future. The key notion we develop is that of routing areas: a family of proof nets which correspond to a fragment of differential linear logic and which implements communication primitives. We develop routing areas as a parametrizable device and study their theory. We then illustrate their expressivity by translating a concurrent λ-calculus featuring concurrency, references and replication to a fragment of differential nets. To this purpose, we introduce a language akin to Amadio’s concurrent λ-calculus, but with explicit substitutions for both variables and references. We endow this language with a type and effect system and we prove termination of well-typed terms by a mix of reducibility and a new interactive technique. This intermediate language allows us to prove a simulation and an adequacy theorem for the translation.

Dynamical Analysis incorporates tools from dynamical systems, namely the Transfer Operator, into the framework of Analytic Combinatorics, permitting the analysis of numerous algorithms and objects naturally associated with an underlying dynamical system. This dissertation presents, in the integrated framework of Dynamical Analysis, the probabilistic analysis of seemingly distinct problems in a unified way: the probabilistic study of the recurrence function of Sturmian words, and the probabilistic study of the Continued Logarithm algorithm.

Sturmian words are a fundamental family of words in Word Combinatorics. They are in a precise sense the simplest infinite words that are not eventually periodic. Sturmian words have been well studied over the years, notably by Morse and Hedlund (1940) who demonstrated that they present a notable number theoretical characterization as discrete codings of lines with irrational slope, relating them naturally to dynamical systems, in particular the Euclidean dynamical system. These words have never been studied from a probabilistic perspective. Here, we quantify the recurrence properties of a “random” Sturmian word, which are dictated by the so-called “recurrence function”; we perform a complete asymptotic probabilistic study of this function, quantifying its mean and describing its distribution under two different probabilistic models, which present different virtues: one is a naturaly choice from an algorithmic point of view (but is innovative from the point of view of dynamical analysis), while the other allows a natural quantification of the worst-case growth of the recurrence function. We discuss the relation between these two distinct models and their respective techniques, explaining also how the two seemingly different techniques employed could be linked through the use of the Mellin transform. In this dissertation we also discuss our ongoing work regarding two special families of Sturmian words: those associated with a quadratic irrational slope, and those with a rational slope (not properly Sturmian). Our work seems to show the possibility of a unified study.

The Continued Logarithm Algorithm, introduced by Gosper in Hakmem (1978) as a mutation of classical continued fractions, computes the greatest common divisor of two natural numbers by performing division-like steps involving only binary shifts and substractions. Its worst-case performance was studied recently by Shallit (2016), who showed a precise upper-bound for the number of steps and gave a family of inputs attaining this bound. In this dissertation we employ dynamical analysis to study the average running time of the algorithm, giving precise mathematical constants for the asymptotics, as well as other parameters of interest. The underlying dynamical system is akin to the Euclidean one, and was first studied by Chan (around 2005) from an ergodic, but the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying this system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the algorithm by Dynamical Analysis.