I have the main idea, yet I'm uncertain on how to construct this PDA (in terms of states, transitions)

We can assume the alphabet $\Sigma$ is {$0,1$}, proving for $\Sigma=${$0,1$} is a sufficient proof that 'covers' all $\Sigma$'s such that $|\Sigma|\geq 2$.

I want to use the fact that the language $L=${$w\in\Sigma^* : w=u_1\circ u_2 , |u_1|=|u_2| , u_1\neq u_2$} can be constructed using a PDA, and that since $u_1\neq u_2$ there must be atleast one character which is in one sub-word, and isnt in the other.


After some playing around, I've constructed 2 PDAs which I believe one might be correct, however I'm not sure how to 'test' them. The PDAs are as such: PDA 1


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Edit 2:

I've managed to construct this PDA (lets call it PDA3): enter image description here

I feel like PDA3 is the right answer, simply because I've tried some examples and all seem to work/fail according to $L$, yet I have no way of formally proving it.

Is there a method to proving? except for experimenting with specific examples.



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