Implicit computational complexity and probabilistic classes

The thesis applies the ICC tecniques to the probabilistic polinomial complexity classes in order to get an implicit characterization of them. The main contribution lays on the implicit characterization of PP (which stands for Probabilistic Polynomial Time) class, showing a syntactical characterisation of PP and a static complexity analyser able to recognise if an imperative program computes in Probabilistic Polynomial Time. The thesis is divided in two parts. The first part focuses on solving the problem by creating a prototype of functional language (a probabilistic variation of lambda calculus with bounded recursion) that is sound and complete respect to Probabilistic Prolynomial Time. The second part, instead, reverses the problem and develops a feasible way to verify if a program, written with a prototype of imperative programming language, is running in Probabilistic polynomial time or not. This thesis would characterise itself as one of the first step for Implicit Computational Complexity over probabilistic classes. There are still open hard problem to investigate and try to solve. There are a lot of theoretical aspects strongly connected with these topics and I expect that in the future there will be wide attention to ICC and probabilistic classes.