GATE | GATE-CS-2007 | Question 30

The language L= {0ⁱ21ⁱ | i≥0 } over the alphabet {0,1, 2} is:
(A) not recursive
(B) is recursive and is a deterministic CFL.
(C) is a regular language.
(D) is not a deterministic CFL but a CFL.


Answer: (B)

Explanation: Let us first design a deterministic pushdown automata for the given language.

• For each occurrence of ‘0’ , we PUSH X in the stack.
• When ‘2’ appears, no stack operation is performed. But, state of the automata is changed.
• For each occurrence of ‘1’ , we POP X from the stack.
• If at the end Z₀ is on the stack top then input string is accepted

We also design a Turing machine for the given language.

• When ‘0’ appears in the input string , we replace it with X .Then, traverse to the rightmost corner and replace ‘1’ with Y.
• We go back to the leftmost ‘0’ and repeat the above process.
• While traversing rightwards from the beginning of the input string, if after X, ‘2’ appears and after ‘2’, Y appears then we reach the HALT state.

  Thus, the given language is recursive. Every recursive language is a CFL.
  Thus, option (B) is the answer.

  Please comment below if you find anything wrong in the above post.

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