Probabilistic causes in Markov chains

By combining two of the central paradigms of causality, namely counterfactual reasoning and probability-raising, we introduce a probabilistic notion of cause in Markov chains. Such a cause consists of finite executions of the probabilistic system after which the probability of an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-regular effect exceeds a given threshold. The cause, as a set of executions, then has to cover all behaviors exhibiting the effect. With these properties, such causes can be used for monitoring purposes where the aim is to detect faulty behavior before it actually occurs. In order to choose which cause should be computed, we introduce multiple types of costs to capture the consumption of resources by the system or monitor from different perspectives, and study the complexity of computing cost-minimal causes.