Static Analysis of Programs with Imprecise Probabilistic Inputs

International audienceHaving a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.Il est important de disposer d'une abstraction précise mais correcte des entrées d'un programme numérique pour analyser ses comportements. Pour de nombreux programmes, ces entrées sont probabilistes, mais la distribution réellement utilisée n'est connue que partiellement. Nous présentons un cadre d'analyse statique permettant le raisonnement sur des programmes dont les entrées sont données sous forme de probabilités imprécises: nous définissons une sémantique collectrice fondée sur la notion de prévisions et une sémantique abstraite fondée sur une extension des structures de Dempster-Shafer. Nous démontrons la correction de notre approche et montrons sur des exemples réalistes le genre d'invariants que nous sommes capables d'inférer

Approximating Markov Processes by Averaging

We recast the theory of labelled Markov processes in a new setting, in a way "dual" to the usual point of view. Instead of considering state transitions as a collection of subprobability distributions on the state space, we view them as transformers of real-valued functions. By generalizing the operation of conditional expectation, we build a category consisting of labelled Markov processes viewed as a collection of operators; the arrows of this category behave as projections on a smaller state space. We define a notion of equivalence for such processes, called bisimulation, which is closely linked to the usual definition for probabilistic processes. We show that we can categorically construct the smallest bisimilar process, and that this smallest object is linked to a well-known modal logic. We also expose an approximation scheme based on this logic, where the state space of the approximants is finite; furthermore, we show that these finite approximants categorically converge to the smallest bisimilar process.Nous reconsidérons les processus de Markov étiquetés sous une nouvelle approche, dans un certain sens "dual'' au point de vue usuel. Au lieu de considérer les transitions d'état en état en tant qu'une collection de distributions de sous-probabilités sur l'espace d'états, nous les regardons en tant que transformations de fonctions réelles. En généralisant l'opération d'espérance conditionelle, nous construisons une catégorie où les objets sont des processus de Markov étiquetés regardés en tant qu'un rassemblement d'opérateurs; les flèches de cette catégorie se comportent comme des projections sur un espace d'états plus petit. Nous définissons une notion d'équivalence pour de tels processus, que l'on appelle bisimulation, qui est intimement liée avec la définition usuelle pour les processus probabilistes. Nous démontrons que nous pouvons construire, d'une manière catégorique, le plus petit processus bisimilaire à un processus donné, et que ce plus petit object est lié à une logique modale bien connue. Nous développons une méthode d'approximation basée sur cette logique, où l'espace d'états des processus approximatifs est fini; de plus, nous démontrons que ces processus approximatifs convergent, d'une manière catégorique, au plus petit processus bisimilaire

Reasoning with random sets: An agenda for the future

In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.Comment: 94 pages, 17 figure

Mixed powerdomains for probability and nondeterminism

We consider mixed powerdomains combining ordinary nondeterminism and probabilistic nondeterminism. We characterise them as free algebras for suitable (in)equation-al theories; we establish functional representation theorems; and we show equivalencies between state transformers and appropriately healthy predicate transformers. The extended nonnegative reals serve as `truth-values'. As usual with powerdomains, everything comes in three flavours: lower, upper, and order-convex. The powerdomains are suitable convex sets of subprobability valuations, corresponding to resolving nondeterministic choice before probabilistic choice. Algebraically this corresponds to the probabilistic choice operator distributing over the nondeterministic choice operator. (An alternative approach to combining the two forms of nondeterminism would be to resolve probabilistic choice first, arriving at a domain-theoretic version of random sets. However, as we also show, the algebraic approach then runs into difficulties.) Rather than working directly with valuations, we take a domain-theoretic functional-analytic approach, employing domain-theoretic abstract convex sets called Kegelspitzen; these are equivalent to the abstract probabilistic algebras of Graham and Jones, but are more convenient to work with. So we define power Kegelspitzen, and consider free algebras, functional representations, and predicate transformers. To do so we make use of previous work on domain-theoretic cones (d-cones), with the bridge between the two of them being provided by a free d-cone construction on Kegelspitzen