GATE CS 2025 Set 2 — Question 25
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Theory of Computation → Context-Free Languages → Decision Problems (CFL)
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Question
Let be Context Free Grammars (CFGs) and be a regular expression. For a grammar , let
L(G) denote the language generated by .Which ONE among the following questions is decidable?Correct answer
(D) Is L(G₁) = ∅ ?
Solution
We analyze the decidability of each problem for Context Free Languages (CFLs):
1.Equivalence of CFGs (): This is undecidable. There is no algorithm to determine if two arbitrary CFGs generate the same language.
2.Intersection Emptiness (): This is undecidable. The intersection of two CFLs is not necessarily a CFL, and determining if their intersection is empty is undecidable (related to the Post Correspondence Problem).
3.Equality to a Regular Language (): This is undecidable. Specifically, the problem of determining if (Universality) is undecidable. Since is a regular language, checking is undecidable in the general case.
4.Emptiness (): This is decidable. We can determine if the start symbol of the grammar can generate any string of terminals by checking reachability in the grammar (e.g., marking productive symbols).
Therefore, only option (D) is decidable.Continue learning with Success Tracker
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