A Fixpoint Logic for Labeled Markov Processes

We develop in this abstract a probabilistic fixpoint logic for Labeled Markov Processes (LMPs). One reason for doing this comes from [2, 3]. There, it was shown that the LMP logic characterizing bisimulation can be used to define in a natural way finite-state approximants of LMPs. An extension of this logic with fixpoints such as the one we propose allows for stronger notions of approximants. Steady properties, i.e. properties related to infinite behaviours, can be obtained in finite approximations. Our logic only deals with greatest fixpoints. In a probabilistic setting, one has a the pending constraint that all logical terms denote measurable sets. Since measurability is preserved by countable boolean operations, only continuous or cocontinuous operators are meaningful. Least fixpoints are trivial in our case (more about this below) so we’re left with greatest fixpoints. As an illustration of the descriptive power of the logic, we provide an explicit construction of the coarsest probabilistic simulation of a given finite LMP by an arbitrary one. This construction is interesting in its own right. Finally a continuous state space example is given. An LMP can be described as a family of probabilities (p(s))s∈S indexed by the state space S, p(s)(A) representing the probability that the process will jump from s to A a measurable subset of S. In some special circumstances (when all the p(s) are mutually absolutely continuous, i.e. define the same negligible events) the Radon-Nikodým theorem makes it possible to extend the σ-algebra of events into a complete boolean algebra and therefore a logic with both fixpoints and arbitrary monotone operators seems possible. We might pursue this option in the future.