Krishnendu Chatterjee, Koushik Sen, and Thomas A. Henzinger
We study the problem of model checking Interval-valued Discrete-Time Markov Chains (IDTMCs). IDTMCs are discrete-time finite Markov Chains for which the exact transition probabilities are not known. Instead, in IDTMCs, each transition is associated with an interval in which the actual transition probability must lie. We consider two semantic interpretations for the uncertainty in the transition probabilities of an IDTMC. In the first interpretation, we think of the IDTMC as representing a (possibly uncountable) family of (classical) discrete-time Markov chains, where each member of the family is a Markov chain whose transition probabilities lie within the interval range given in the IDTMC. We call this semantic interpretation Uncertain Markov Chains (UMC). In the second semantics for the IDTMC, which we call Interval Markov Decision Process (IMDP), we view the uncertainty as being resolved through nondeterminism. In other words, each time a state is visited, an adversary chooses a transition distribution that respects the interval constraints, and we take a probabilistic step according to the chosen distribution.
We introduce the logic omega-PCTL, which can express omega-regular properties including weak and strong fairness (such properties cannot be expressed in PCTL). We show that the omega-PCTL model-checking problem for UMC semantics is decidable in PSPACE (this is the same as the best known upper bound for PCTL), and for IMDP semantics, it is decidable in coNP (this improves the best previously known PSPACE bound for PCTL). We also show that the qualitative fragment of the logic can be solved in coNP for the UMC interpretation, and can be solved in polynomial time for a subclass of IDTMCs with UMC semantics. We also prove lower bounds for these model-checking problems. Finally, we show that the LTL model-checking problem over IDTMCs can be solved for both UMC and IMDP semantics by reduction to the omega-PCTL model-checking problem.
Proceedings of the 11th International Conference on Foundations of Software Science and Computational Structures (FOSSACS), Lecture Notes in Computer Science 4962, Springer, 2008, pp. 302-317.