What is the role of the pumping lemma for context-free languages with epsilon transitions in automata theory?

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What is the role of the pumping lemma for context-free languages with epsilon transitions in automata theory?

The pumping lemma for context-free languages with epsilon transitions is used to prove that a language is not context-free. It states that for any context-free language L, there exists a pumping length p such that any string s in L with length greater than or equal to p can be divided into five parts: uvxyz, satisfying certain conditions. By repeatedly pumping the substring v and y, we can generate strings that are not in L, thus proving that L is not context-free.