Infinite-state Stochastic and Parameterized Systems

A major current challenge consists in extending formal methods in order to handle infinite-state systems. Infiniteness stems from the fact that the system operates on unbounded data structure such as stacks, queues, clocks, integers; as well as parameterization. Systems with unbounded data structure are natural models for reasoning about communication protocols, concurrent programs, real-time systems, etc. While parameterized systems are more suitable if the system consists of an arbitrary number of identical processes which is the case for cache coherence protocols, distributed algorithms and so forth. In this thesis, we consider model checking problems for certain fundamental classes of probabilistic infinite-state systems, as well as the verification of safety properties in parameterized systems. First, we consider probabilistic systems with unbounded data structures. In particular, we study probabilistic extensions of Lossy Channel Systems (PLCS), Vector addition Systems with States (PVASS) and Noisy Turing Machine (PNTM). We show how we can describe the semantics of such models by infinite-state Markov chains; and then define certain abstract properties, which allow model checking several qualitative and quantitative problems. Then, we consider parameterized systems and provide a method which allows checking safety for several classes that differ in the topologies (linear or tree) and the semantics (atomic or non-atomic). The method is based on deriving an over-approximation which allows the use of a symbolic backward reachability scheme. For each class, the over-approximation we define guarantees monotonicity of the induced approximate transition system with respect to an appropriate order. This property is convenient in the sense that it preserves upward closedness when computing sets of predecessors.