Identifying the network-wide forwarding behaviors of a packet is essential for many network management applications, including rule verification, policy enforcement, attack detection, traffic engineering, and fault localization. Current tools that can perform packet behavior identification either incur large time and memory costs or do not support real-time updates. In this paper, we present AP Classifier, a control plane tool for packet behavior identification. AP Classifier is developed based on the concept of atomic predicates, which can be used to characterize the forwarding behaviors of packets. Experiments using the data plane network state of two real networks show that the processing speed of AP Classifier is faster than existing tools by at least an order of magnitude. Furthermore, AP Classifier uses very small memory and is able to support real-time updates.