Stochastic analogues of invariants martingales in stochastic Event-B

In conventional formal model based development frameworks, invariants play a key role in controlling the behaviour of the model (when they contribute to the definition of the model) or in verifying the model's properties (when the model, independently defined, is required to preserve the invariants). However, when variables take values distributed according to some probability distribution, the possibility of verifying that system behaviour is, in the long term, confined to some acceptable set of states can be severely diminished because the system might, in fact, with low probability fail to be thus confined. This short paper proposes martingales as suitable analogues of invariants for capturing suitable properties of non-terminating systems whose behaviour is with high probability good, yet where a small chance of poor behaviour remains. The idea is explored in the context of the well-known Event-B framework.