We investigate a finer-grained understanding of the characteristics of particular deterministic finite automata (DFA). Specifically, we study and identify the transitions of a DFA that are more important for maintaining the correctness of the underlying regular language associated with this DFA. To estimate transition importance, we develop an approach that is similar to the approach widely used to expose the vulnerability of neural networks with the adversarial example problem. In our approach, we propose an adversarial model that reveals the sensitive transitions embedded in a DFA. In addition, we find for a DFA its critical patterns where a pattern is a substring that can be taken as the signature of this DFA. Our defined patterns can be implemented as synchronizing words, which represent the passages from different states to the absorbing state of a DFA. Finally, we validate our study through empirical evaluations by showing that our proposed algorithms can effectively identify important transitions and critical patterns. To our knowledge, this is some of the first work to explore adversarial models for DFAs and is important due to the wide use of DFAs in cyberphysical systems.