Solution

The correct answer is {anb3n|n > 0} 

Explanation:

To determine the language accepted by the given Pushdown Automaton (PDA), we need to analyze its transitions and the behavior of its stack.

To determine the language accepted by the given Pushdown Automaton (PDA), we need to analyze its transitions and the behavior of its stack.

The PDA transitions are given as follows:

δ(q0, a, z0) = (q0, aaaz0)

δ(q0, a, a) = (q0, aaaa)

δ(q0, b, a) = (q1, λ)

δ(q1, b, a) = (q1, λ)

δ(q1, ∈, z0) = (qf, z0), where qf is a final state. 

Let's interpret these transitions:

• From q0, upon seeing an a with z0 on the stack, it pushes three a's onto the stack.
• From q0, upon seeing an a with a on the stack, it pushes three a's onto the stack.
• From q0, upon seeing a b with a on the stack, it removes a from the stack (using \( \lambda\)).
• From q1, upon seeing a b with a on the stack, it continues to remove a from the stack.
• When q1 reaches \(\epsilon\) (empty input) with z0 on the stack, it transitions to qf and empties the stack.

Now, let's analyze the language recognized by this PDA:

• The PDA starts in state q0 with z0 on the stack.
• For every a read in state q0, it adds aaa to the stack, depending on whether z0 or a is on the stack initially.
• For every b encountered, it removes an a from the stack.
• When no input is left (\(\epsilon\)), if z0 is on the stack, it accepts (moves to qf).

The key observation here is the balanced nature of the stack operations:

• For each a, the stack grows by three a's.
• For each b, the stack decreases by one a.

The language accepted by the PDA is . {anb3n|n > 0} 

Therefore, the correct answer is: {anb3n|n > 0}