Online Monitoring $ω$-Regular Properties in Unknown Markov Chains

We study runtime monitoring of $ω$-regular properties. We consider a simple setting in which a run of an unknown finite-state Markov chain $\mathcal M$ is monitored against a fixed but arbitrary $ω$-regular specification $φ$. The purpose of monitoring is to keep aborting runs that are "unlikely" to satisfy the specification until $\mathcal M$ executes a correct run. We design controllers for the reset action that (assuming that $φ$ has positive probability) satisfy the following property w.p.1: the number of resets is finite, and the run executed by $\mathcal M$ after the last reset satisfies $φ$.

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