# PDA that accepts if there is an “a” in the second half of the string

How could I build a PDA on $$Σ = \{a,b\}$$ that accepts if there is an "a" in the second half of the string?

Formally the language is the following: $$L = \{xy \mid x ∈ Σ^*, y ∈ Σ^*aΣ^* \land |x| ≥ |y|\}$$.

The PDA has three phases. In the first phase, it adds $$A$$ to the stack per character. It switches to the second phase nondeterministically. In the second phase, it removes $$A$$ per character. It switches to the third phase nondeterministically, during which it empties the stack. The PDA accepts if it encountered an $$a$$ during the second phase.