# Is there any property about height of PDA?

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.

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)