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I am learning about context free languages.

I understand how $\{a^nb^nc^n|n>0\}$ can be shown to be not context free using the pumping lemma for CFL's.

Intuitively however it seems that a pushdown automata to recognize $\{a^nb^nc^n|n>0\}$ can be constructed. This PDA would initially push single a's into its stack whenever it sees an a in the input. It would change state when it first encounters a b and do nothing on the stack for every b it encounters. When the first c is encountered, it would again change state and pop single a's for every c encountered. If the stack is empty at the end of the input the language is recognized as $\{a^nb^nc^n|n>0\}$.

There must be something I am overlooking whilst constructing the PDA as a language is context free if its has a PDA recognizing it. Please point out my mistake.

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    $\begingroup$ You are required to match the number of $a$, $b$, and $c$. If you simply ignore the $b$'s then you cannot guarantee that all type of symbols have the same length. $\endgroup$
    – Russel
    Oct 31, 2022 at 13:00

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Because there is no control on the number of $b$'s, such a pushdown automaton would recognize $\{a^nb^mc^n\mid n, m> 0\}$ which is indeed context-free.

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