In this paper we extend the predicate logic introduced by Beauquier et al. in order to deal with Markov decision processes. We prove that with respect to qualitative probabilistic properties, model checking is decidable for this logic applied to Markov decision processes. Furthermore we apply our logic to probabilistic timed transition systems using predicates on clocks. We prove that results on Markov decision processes hold also for probabilistic timed transition systems. The interest of this logic lies in particular on the fact that some important properties are expressible in this logic but not expressible in pCTL.

A predicate logic of probability, close to the logics of probability of Halpern et al., is introduced. Our main result concerns the following model-checking problem: deciding whether a given formula holds on the structure defined by a given finite probabilistic process. We show that this model-checking problem is decidable for a rather large subclass of formulas of a second-order monadic logic of probability. We discuss also the decidability of satisfiability and compare our logic of probability with the probabilistic temporal logic pCTL*. * Partially supported by French-Israeli Arc-en-ciel/Keshet project No. 30 and No. 15.

A major result concerning Temporal Logics (T L) is Kamp's theorem which states that the temporal logic over the pair of modalities X until YandXsinceY is expressively complete for the first‐order fragment of monadic logic of order over the natural numbers. We show that there is no finite set of modalities B such that the temporal logic over B and monadic logic of order have the same expressive power over the natural numbers. As a consequence of our proof, we obtain that there is no finite base temporal logic which is expressively complete for the μ‐calculus.

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