We survey various notions of probabilistic automata and probabilistic bisimulation, accumulating in an expressiveness hierarchy of probabilistic system types. The aim of this paper is twofold: On the one hand it provides an overview of existing types of probabilistic systems and, on the other hand, it explains the relationship between these models.

Probability, be it inherent or explicitly introduced, has become an important issue in the verification of programs. In this paper we study a formalism which allows reasoning about programs which can act probabilistically. To describe probabilistic programs, a basic programming language with an operator for probabilistic choice is introduced and a denotational semantics is given for this language. To specify properties of probabilistic programs, standard first order logic predicates are insufficient, so a notion of probabilistic predicates is introduced. A Hoare-style proof system to check properties of probabilistic programs is given. The proof system for a sublanguage is shown to be sound and complete; the properties that can be derived are exactly the valid properties. Finally some typical examples illustrate the use of the probabilistic predicates and the proof system.

In this paper we present an algebraic language for the specification of probabilistic and nondeterministic processes, PNAL, which is a probabilistic extension of EPL (Algebraic Theory of Processes, M. Hennessy) that maintains nondeterminism.

For a process language, featuring nondeterministic and probabilistic choice, a parallel operator and a failure construct, a notion of bisimulation is proposed. As one can interpret recovery from failure with respect to nondeterministic and probabilistic choice in various ways, a single transition system gives rise to several operational models. A uniform way to abstract the `first steps' underlies, for each of these models, the definition of the proposed bisimulation. This bisimulation specializes to Park-Milner bisimulation for the non-probabilistic fragment of the language on the one hand, and to Larsen-Skou bisimulation for the deterministic/probabilistic part of the language, on the other hand. Furthermore, a conditional congruence result is obtained. 1 Introduction Understanding the interplay of nondeterminacy and probability is a key issue in the development of formal techniques for specification and validation of probabilistic programs and protocols. In this paper we study a pr...

This paper provides a case-study in the eld of metric semantics for probabilistic programming. Both an operational and a denotational semantics are presented for an abstract process language L pr , which features action refinement and probabilistic choice. The two models are constructed in the setting of complete ultrametric spaces, here based on probability measures of compact support over sequences of actions. It is shown that the standard toolkit for metric semantics works well in the probabilistic context of L pr , e.g. in establishing the correctness of the denotational semantics with respect to the operational one. In addition, it is shown how the method of proving full abstraction -- as proposed recently by the authors for a nondeterministic language with action refinement -- can be adapted to deal with the probabilistic language L pr as well.