Satisfiability Bounds for {ω}-regular Properties in Interval-valued Markov Chains

We derive an algorithm to compute satisfiability bounds for arbitrary {\omega}-regular properties in an Interval-valued Markov Chain (IMC) interpreted in the adversarial sense. IMCs generalize regular Markov Chains by assigning a range of possible values to the transition probabilities between states. In particular, we expand the automata-based theory of {\omega}-regular property verification in Markov Chains to apply it to IMCs. Any {\omega}-regular property can be represented by a Deterministic Rabin Automata (DRA) with acceptance conditions expressed by Rabin pairs. Previous works on Markov Chains have shown that computing the probability of satisfying a given {\omega}-regular property reduces to a reachability problem in the product between the Markov Chain and the corresponding DRA. We similarly define the notion of a product between an IMC and a DRA. Then, we show that in a product IMC, there exists a particular assignment of the transition values that generates a largest set of non-accepting states. Subsequently, we prove that a lower bound is found by solving a reachability problem in that refined version of the original product IMC. We derive a similar approach for computing a satisfiability upper bound in a product IMC with one Rabin pair. For product IMCs with more than one Rabin pair, we establish that computing a satisfiability upper bound is equivalent to lower-bounding the satisfiability of the complement of the original property. A search algorithm for finding the largest accepting and non-accepting sets of states in a product IMC is proposed. Finally, we demonstrate our findings in a case study.