Probabilistic (or quantitative) verification is a branch of formal methods dealing with stochastic models and logic. Probabilistic models capture the behavior of randomized algorithms and other physical systems with certain uncertainty, whereas probabilistic logic expresses the quantitative measure on the probabilistic space defined by the models. Most often, the formal techniques used in studying the behavior of these models and logic are not just mundane extension of its non-probabilistic counterparts. The complexity of these mathematical structures is surprisingly different. The thesis is an effort at improving our continued under- standing of these models and logic. We will begin by looking at few interesting formal representations of discrete stochastic models. Namely, we will address the parameter synthesis problem for parametric linear time temporal logic and model checking of convex Markov decision processes with open intervals. The primary focus of the thesis is however on the satisfiability (or validity) problem of different probabilistic logics. This includes a bounded fragment of probabilistic logic and a simple quantitative (probabilistic) extension of mu-calculus. Decision procedures for the satisfiability problems are developed and a detailed complexity analysis of these problems is provided. The study of automata has been very effective in understanding logic. We will look at the newly conceived notion of p-automata, which are a probabilistic extension of alternating tree automata. As we will see, probabilistic logic exhibits both non-deterministic and stochastic behavior. The semantics of p-automata are extended to capture non-determinism and hence model Markov decision processes
