Let’s define a (P = (Q, \Sigma, \Gamma, \delta, q_0, Z, F)) where:

From (q_0): (\delta(q_0, a, Z) = (q_0, A Z)) # push A for first a (\delta(q_0, a, A) = (q_0, A A)) # push A for each a (\delta(q_0, \epsilon, Z) = (q_f, Z)) # empty string case

Read (a)’s, push (X) for each.

The PDA works as follows:

Phase one was the intake. As the machine read the first block of symbols—the 'a's—it didn't just watch them pass. For every 'a' that entered the system, Elias programmed the PDA to push a marker onto its internal stack. Each 'a' was a debt, a promise of a 'b' to come.

But that’s tricky: if we are in q1, top Z, we know we exhausted a’s. But still b’s may come — we go to q2 and read b’s.

'b's have been read), start pushing the remaining 'b's (e.g., as symbol 'B') onto the stack.