Termination of probabilistic concurrent programs

Abstract

The asynchronous execution behavior of several concurrent processes, which may use randomization, is studied. Viewing each process as a discrete Markov chain over the set of common execution states, we give necessary and sufficient conditions for the processes to converge almost surely to a given set of goal states, under any fair, but otherwise arbitrary schedule, provided that the state space is finite. (These conditions can be checked mechanically.) An interesting feature of the proof method is that it depends only on the topology of the transitions and not on the actual values of the probabilities. We also show that in our model synchronization protocols that use randomization are in certain cases no more powerful than deterministic protocols. This is demonstrated by (a) Proving lower bounds on the size of a shared variable necessary to ensure mutual exlusion and lockout-free behavior of the protocol; and (b) Showing that no fully symmetric 'randomized' protocol can ensure mutual exclusion and freedom from lockout.