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I'm trying to find a PDA for $L$ which modifies the stack height at most one.

$L=\{a^ib^i\mid i\geq 0\}$

I think there is no such PDA but how can I prove it?

My attempt is for a given string, find a relationship between stack height in PDA and leftmost derivation in CFG then prove that there's no CFG which satisfies the condition using pumping lemma. But I couldn't find any relationship. Am I doing right way?

Any help appreciated.

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If you could bound the stack height, say to some constant $c$, then it would have been possible to define an NFA for the task: Simply encode in the NFA states another $c$ values that represent the current values in the stack (by defining $\hat Q=Q\times\Sigma^c$. The rest i will leave to you)

This would contradict the non-regularity of this language, and therefore you cannot give such a bound.

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