Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic

We investigate the decidability of model checking logics of time, knowledge, and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observable discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for the synchronous perfect recall semantics, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second-order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second-order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.