Complexity of Model Checking MDPs against LTL Specifications

Given a Markov Decision Process (MDP) M, an LTL formula \varphi, and a threshold \theta \in [0,1], the verification question is to determine if there is a scheduler with respect to which the executions of M satisfying \varphi have probability greater than (or greater than or equal to) \theta. When \theta = 0, we call it the qualitative verification problem, and when \theta \in (0,1], we call it the quantitative verification problem. In this paper we study the precise complexity of these problems when the specification is constrained to be in different fragments of LTL.