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Given a Markov Decision Process (MDP) M, an LTL formula \varphi, and a threshold \theta \in [0,1], the verification question is to determine if there is a scheduler with respect to which the executions of M satisfying \varphi have probability greater than (or greater than or equal to) \theta. When \theta = 0, we call it the qualitative verification problem, and when \theta \in (0,1], we call it the quantitative verification problem. In this paper we study the precise complexity of these problems when the specification is constrained to be in different fragments of LTL.

A probabilistic finite automaton (PFA) A is said to be regular-approximable with respect to (x,y), if there is a regular language that contains all words accepted by A with probability at least x+y, but does not contain any word accepted with probability at most x. We show that the problem of determining if a PFA A is regular-approximable with respect to (x,y) is not recursively enumerable. We then show that many tractable sub-classes of PFAs identified in the literature - hierarchical PFAs, polynomially ambiguous PFAs, and eventually weakly ergodic PFAs - are regular-approximable with respect to all (x,y). Establishing the regular-approximability of a PFA has the nice consequence that its value can be effectively approximated, and the emptiness problem can be decided under the assumption of isolation.

We investigate how the semantics of a hybrid automaton deviates with respect to syntactic perturbations on the hybrid automaton. We consider syntactic perturbations of a hybrid automaton, wherein the syntactic representations of its elements, namely, initial sets, invariants, guards, and flows, in some logic are perturbed. Our main result establishes a continuity like property that states that small perturbations in the syntax lead to small perturbations in the semantics. More precisely, we show that for every real number epsilon>0 and natural number k, there is a real number delta>0 such that H^delta, the delta syntactic perturbation of a hybrid automaton H, is epsilon-simulation equivalent to H up to k transition steps. As a byproduct, we obtain a proof that a bounded safety verification tool such as dReach will eventually prove the safety of a safe hybrid automaton design (when only non-strict inequalities are used in all constraints) if dReach iteratively reduces the syntactic parameter delta that is used in checking approximate satisfiability. This has an immediate application in counter-example validation in a CEGAR framework, namely, when a counter-example is spurious, then we have a complete procedure for deducing the same.

International audience; Book Front Matter of LNCS 9039