  The paper studies a probabilistic notion of causes in Markov chains that
relies on the counterfactuality principle and the probability-raising property.
This notion is motivated by the use of causes for monitoring purposes where the
aim is to detect faulty or undesired behaviours before they actually occur. A
cause is a set of finite executions of the system after which the probability
of the effect exceeds a given threshold. We introduce multiple types of costs
that capture the consumption of resources from different perspectives, and
study the complexity of computing cost-minimal causes.
