Libraries usually impose constraints on how clients should use them. Often these constraints are not well-documented. In this paper, we address the problem of recovering such constraints automatically, a problem referred to as speciﬁcation mining. Given some client programs that use a given library, we identify constraints on the library usage that are (almost) satisﬁed by the given set of clients. The class of rules we target for mining combines simple binary temporal operators with state predicates (involving equality constraints) and quantiﬁcation. This is a simple yet expressive subclass of temporal properties that allows us to capture many common API usage rules. We focus on recovering rules from execution traces and apply classical data mining concepts to be robust against bugs (API usage rule violations) in clients. We present new algorithms for mining rules from execution traces. We show how a propositional rule mining algorithm can be generalized to treat quantiﬁcation and state predicates in a uniﬁed way. Our approach enables the miner to be complete — mine all rules within the targeted class that are satisﬁed by the given traces — while avoiding an exponential blowup. We have implemented these algorithms and used them to mine API usage rules for several Windows APIs. Our experiments show the efﬁciency and effectiveness of our approach.