Modeling and verification of randomized distributed real-time systems

Randomization is an excellent tool for the design of distributed algorithms, sometimes yielding efficient solutions to problems that are inherently complex, or even unsolvable, in the setting of deterministic algorithms. However, this tool has a price: even simple randomized algorithms can be extremely hard to verify and analyze. This thesis addresses the problem of verification of randomized distributed algorithms. We consider the problem both from the theoretical and the practical perspective. Our theoretical work builds a new mathematical model of randomized distributed computation; our practical work develops techniques to be used for the actual verification of randomized systems. Our analysis involves both untimed and timed systems, so that real-time properties can be investigated. Our model for randomized distributed computation is an extension of labeled transition systems. A Probalistic automation is a state machine with transitions, where, unlike for labeled transitions systems, a transition from a state leads to a discrete probability distribution over pairs consisting of a label and a state, rather than to a single label and a single state. A probalistic automation contains pure nondeterministic behavior since from each state there can be several transisitions, and probalistic behavior since once a transaction is chosen the label that occurs and the state that is reached are determined by a probability distribution. The resolution of pure nondeterminism leads to probalistic executions, which are Markov chain like structures. Once the pure nondeterminism is resolved, the probalistic behavior of a probalistic automation can be studied. The properties of a randomized algorithm are stated in terms of satisfying some other property with a minimal or maximal probability no matter how the nondeterminism is resolved. In stating the properties of an algorithm we also account for the possibility of imposing restrictions on the ways in which the the nondeterminism is resolved (e.g., fair scheduling, oblivious scheduling,...). We develop techniques to prove the correctness of some property by reducing the problem to the verification of properties of non-randomized systems. One technique is based on coin lemmas, which state the lower bounds on the probability that some chosen random draws give some chosen outcomes no matter how the nondeterminism is resolved. We identify a collection of progress statements which can be used to prove upper bounds to the expected running time of an algorithm. The methods are applied to prove that the randomized dining philosophers algorithm of Lehmann and Rabin guarantees progress in expected constant time and that the randomized algorithm for agreement of Ben-Or guarantees agreement in expected exponential time. To ensure that our new model has strong mathematical foundations, we extend some of the common semantics for labeled transitions systems to the probalistic framework. We define a compositional trace semantic