Parametric LTL on Markov Chains

This paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the untilmodality is equipped with a bound that contains variables; e.g., ♦6x φ asserts that φ holds within x time steps, where x is a variable on natural numbers. The central problem studied in this paper is to determine the set of parameter valuations V≺p(φ) for which the probability to satisfy pLTL-formula φ in a Markov chain meets a given threshold ≺ p, where ≺ is a comparison on reals and p a probability. As for pLTL determining the emptiness of V>0(φ) is undecidable, we consider several logic fragments. We consider parametric reachability properties, a sub-logic of pLTL restricted to next and ♦6x, parametric Büchi properties and finally, a maximal subclass of pLTL for which emptiness of V>0(φ) is decidable.