Controller Synthesis and Verification for Markov Decision Processes with Qualitative Branching Time Objectives

We show that the controller synthesis and verification problems for Markov decision processes with qualitative PECTL∗ objectives are 2-EXPTIME complete. More precisely, the algorithms are polynomial in the size of a given Markov decision process and doubly exponential in the size of a given qualitative PECTL∗ formula. Moreover, we show that if a given qualitative PECTL∗ objective is achievable by some strategy, then it is also achievable by an effectively constructible one-counter strategy, where the associated complexity bounds are essentially the same as above. For the fragment of qualitative PCTL objectives, we obtain EXPTIME completeness and the algorithms are only singly exponential in the size of the formula.