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This paper focuses on optimizing probabilities of events of interest definedover general controlled discrete-time Markov processes. It is shown that theoptimization over a wide class of $\omega$-regular properties can be reduced tothe solution of one of two fundamental problems: reachability and repeatedreachability. We provide a comprehensive study of the former problem and aninitial characterisation of the (much more involved) latter problem. A casestudy elucidates concepts and techniques.

Quantitative model-checking of controlled discrete-time Markov processes