Conditional Probabilities over Probabilistic and Nondeterministic Systems

This paper introduces the logic cpCTL, which extends the probabilistic temporal logic pCTL with conditional probability, allowing one to express that the probability that φ is true given that ψ is true is at least a. We interpret cpCTL over Markov Chain and Markov Decision Processes. While model checking cpCTL over Markov Chains can be done with existing techniques, those techniques do not carry over to Markov Decision Processes. We present a model checking algorithm for Markov Decision Processes. We also study the class of schedulers that suffice to find the maximum and minimum probability that φ is true given that ψ is true. Finally, we present the notion of counterexamples for cpCTL model checking and provide a method for counterexample generation.