Pattern matching over event streams is increasingly being employed in many areas including financial services, RFIDbased inventory management, click stream analysis, and electronic health systems. While regular expression matching is well studied, pattern matching over streams presents two new challenges: Languages for pattern matching over streams are significantly richer than languages for regular expression matching. Furthermore, efficient evaluation of these pattern queries over streams requires new algorithms and optimizations: the conventional wisdom for stream query processing (i.e., using selection-join-aggregation) is inadequate. In this paper, we present a formal evaluation model that offers precise semantics for this new class of queries and a query evaluation framework permitting optimizations in a principled way. We further analyze the runtime complexity of query evaluation using this model and develop a suite of techniques that improve runtime efficiency by exploiting sh...