Let Ξ£ = {π, π, π, π} and let πΏ = {π ππ π π ππ β β£ π,π, π, β β₯β¦
2026
Let Ξ£ = {π, π, π, π} and let πΏ = {π ππ π π ππ β β£ π,π, π, β β₯ 0}. Which of the following constraints ensure(s) that the language πΏ is context-free?
- A.
i + π = π + β
- B.
π = π and π = β
- C.
i= β and π = k
- D.
π + π = π + β
Attempted by 12 students.
Show answer & explanation
Correct answer: A, C, D
Analysis of Constraints
The base language L = {a^i b^j c^k d^β | i, j, k, β β₯ 0} is regular. The context-free status depends on the constraints linking the counts.
Option A: i + k = j + β
This constraint implies i - j = β - k. A PDA can push for 'a' and 'c', and pop for 'b' and 'd'. State transitions handle stack underflow, making this language context-free.
Option B: i = k and j = β
This requires matching 'a' with 'c' and 'b' with 'd' independently. The structure a^i b^j c^i d^j is not context-free because the stack cannot track two independent counts separated by other symbols.
Option C: i = β and j = k
This forms a^i b^j c^j d^i. A PDA pushes 'a' and 'b', pops 'b' for 'c', and pops 'a' for 'd'. The nesting is valid, so this language is context-free.
Option D: i + j = k + β
This balances the first half against the second half. A PDA pushes for 'a' and 'b', then pops for 'c' and 'd'. The linear relationship allows verification with a single stack.
Conclusion
Options A, C, and D ensure the language is context-free. Option B does not.