Université Paris Ix Dauphine

The study of probabilistic programs is of considerable interest for the validation of networking protocols, embedded systems, or simply for compiling optimizations. It is also a difficult matter, due to the undecidability of properties on infinite-state deterministic programs, as well as the difficulties arising from probabilistic aspects. In this thesis, we propose a formulaic language for the specification of trace properties of probabilistic, nondeterministic transition systems, encompassing those that can be specified using deterministic Büchi automata. Those properties are in general undecidable on infinite processes. This language has both a concrete semantics in terms of sets of traces, as well as an abstract semantics in terms of measurable functions. We then apply abstract interpretation-based techniques to give upper bounds on the worst-case probability of the studied property. We propose an enhancement of this technique when the state space is partitioned — for instance along the program points —, allowing the use of faster iteration methods. We propose two abstract domains suitable for this analysis, one parameterized by an abstract domain suitable for nondeterministic (but not probabilistic) abstract interpretation, one modeling extended normal distributions. An alternative method to get such upper bounds works is to apply forward abstract interpretation on measures. We propose two abstract domains suitable for this analysis, one parameterized by an abstract domain suitable for nondeterministic abstract interpretation, one modeling sub-exponential queues. This latter domain allows proving probabilistic termination of programs. The methods described so far are symbolic and do not make use of the statistical properties of probabilities. On the other hand, a well-known way to obtain informations on probabilistic distributions is the Monte-Carlo method. We propose an abstract Monte-Carlo method featuring randomized abstract interpreters.