Parametric LTL on Markov Chains

Abstract

Part 2: Track B: Logic, Semantics, Specification and VerificationInternational audienceThis paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the until-modality is equipped with a bound that contains variables; e.g., $\Diamond _{\leqslant x}\ \varphi$ 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 $\Diamond _{\leqslant x}$, parametric Büchi properties and finally, a maximal subclass of pLTL for which emptiness of V> 0(ϕ) is decidable