Abstract. We introduce a symbolic model checking procedure for Probabilistic Computation Tree Logic PCTL over labelled Markov chains as models. Model checking for probabilistic logics typically involves solving linear equation systems in order to ascertain the probability of a given formula holding in a state. Our algorithm is based on the idea of representing the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al [14]. Our procedure, based on the algorithm used by Hansson and Jonsson [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus [9]. 1

It is known that, in both asynchronous and synchronous networks, no Byzantine Agreement (BA) protocol for n players exists if d e of the players are faulty (in other words, no BA protocol is d e-resilient). The only known asynchronous (d e \Gamma 1)-resilient BA protocol runs in expected exponential time, and the best resilience achieved by an asynchronous protocol with polynomial complexity is (d 4 e \Gamma 1). The question whether there exists an asynchronous (d BA protocol with polynomial complexity remained open.

Abstract. We consider the timed automata model of [3], which allows the analysis of real-time systems expressed in terms of quantitative timing constraints. Traditional approaches to real-time system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for real-time systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, “with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2”. 1

Bisimulations that abstract from internal computation have proven to be useful for verification of compositionally defined transition systems. In the literature of probabilistic extensions of such transition systems, similar bisimulations are rare. In this paper, we introduce weak and branching bisimulation for fully probabilistic systems, transition systems where nondeterministic branching is replaced by probabilistic branching. In contrast to the nondeterministic case, both relations coincide. We give an algorithm to decide weak (and branching) bisimulation with a time complexity cubic in the number of states of the fully probabilistic system. This meets the worst case complexity for deciding branching bisimulation in the nondeterministic case. In addition, the relation is shown to be a congruence with respect to the operators of PLSCCS , a lazy synchronous probabilistic variant of CCS. We illustrate that due to these properties, weak bisimulation provides all the crucial ingredients...

This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems --- like large-scale computers, client-server architectures, networks --- can accurately be described using such stochastic specification formalisms.

In this thesis, we present ecient implementation techniques for probabilistic model checking, a method which can be used to analyse probabilistic systems such as randomised distributed algorithms, fault-tolerant processes and communication networks. A probabilistic model checker inputs a probabilistic model and a speci cation, such as \the message will be delivered with probability 1", \the probability of shutdown occurring is at most 0.02" or \the probability of a leader being elected within 5 rounds is at least 0.98", and can automatically verify if the speci cation is true in the model.

A major reason for studying probabilistic processes is to establish a link between a formal model for describing functional system behaviour and a stochastic process. Compositionality is an essential ingredient for specifying systems. Parallel composition in a probabilistic setting is complicated since it gives rise to non-determinism, for instance due to interleaving of independent autonomous activities. This paper presents a detailed study of the resolution of non-determinism in an asynchronous generative setting. Based on the intuition behind the synchronous probabilistic calculus PCCS we formulate two criteria that an asynchronous parallel composition should fulfill. We provide novel probabilistic variants of parallel composition for CCS and CSP and show that these operators satisfy these general criteria, opposed to most existing proposals. Probabilistic bisimulation is shown to be a congruence for these operators and their expansion is addressed.

Long-run average properties of probabilistic systems refer to the average behavior of the system, measured over a period of time whose length diverges to infinity. These properties include many relevant performance and reliability indices, such as system throughput, average response time, and mean time between failures. In this paper, we argue that current formal specification methods cannot be used to specify long-run average properties of probabilistic systems. To enable the specification of these properties, we propose an approach based on the concept of experiments. Experiments are labeled graphs that can be used to describe behavior patterns of interest, such as the request for a resource followed by either a grant or a rejection. Experiments are meant to be performed infinitely often, and it is possible to specify their long-run average outcome or duration. We propose simple extensions of temporal logics based on experiments, and we present model-checking algorithms for the verif...

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Abstract. Anonymity means that the identity of the user performing a certain action is maintained secret. The protocols for ensuring anonymity often use random mechanisms which can be described probabilistically. In this paper we propose a notion of weak probabilistic anonymity, where weak refers to the fact that some amount of probabilistic information may be revealed by the protocol. This information can be used by an observer to infer the likeliness that the action has been performed by a certain user. The aim of this work is to study the degree of anonymity that the protocol can still ensure, despite the leakage of information. We illustrate our ideas by using the example of the dining cryptographers with biased coins. We consider both the cases of nondeterministic and probabilistic users. Correspondingly, we propose two notions of weak anonymity and we investigate their respective dependencies on the biased factor of the coins. 1