Model Checking Almost All Paths Can Be Less Expensive Than Checking All Paths

We compare the complexities of the following two model checking problems: checking whether a linear-time formula is satis ed by all paths (which we call universal model checking) and checking whether a formula is satis ed by almost all paths (which we call fair model checking here). For many interesting classes of linear-time formulas, both problems have the same complexity: for instance, they are PSPACE-complete for LTL. In this paper, we show that fair model checking can have lower complexity than universal model checking, viz., we prove that fair model checking for L(F∞) can be done in time linear in the size of the formula and of the system, while it is known that universal model checking for L(F∞) is co-NP-complete. L(F∞) denotes the class of LTL formulas in which F∞ is the only temporal operator. We also present other new results on the complexity of fair and universal model checking. In particular, we prove that fair model checking for RLTL is co-NP-complete.