Solution

The correct answer is A & B Only.

Key Points

• Pushdown Automata (PDA) accept exactly the class of Context-Free Languages (CFL).
• Languages that require more than one counter or are non-context-free cannot be accepted by PDA.

Step-by-Step Solution

• A. L = {aⁿbⁿcⁿ}
  This language requires counting three equal numbers simultaneously. One stack can match only two counts. Hence it is not context-free. PDA cannot accept it.
• B. L = {aᵖ : p is prime}
  This language is not context-free (number theory property). PDA cannot check primality. Hence not accepted by PDA.
• C. L = {wcwᴿ}
  Palindrome with middle symbol. PDA can push w and match reverse using stack. Context-free → Accepted.
• D. L = {w ∈ {a,b}* : nₐ(w) = n_b(w)}
  Equal number of a’s and b’s is a context-free language. PDA can count difference using stack.
• E. L = {aⁿbᵐcⁿ⁺ᵐ}
  Using stack, PDA counts n and m then matches c’s. This is context-free → Accepted.

Additional Information

• PDA uses one stack → behaves like one counter.
• Languages needing two independent counters are not CFL.