Which one of the following problems is undecidable? A) Deciding if a given context-free grammar is ambiguous. B) Deciding if a given string is generated by a given context-free grammar. C) Deciding if the language generated by a given context-free grammar is empty. D) Deciding if the language generated by a given context-free grammar is finite.

Official answer: A) Deciding if a given context-free grammar is ambiguous.. Explanation: Let us analyze the decision problems for Context-Free Grammars (CFGs) / Context-Free Languages (CFLs): - (A) Ambiguity of a CFG: Deciding whether an arbitrary CFG is ambiguous is undecidable (proven via reduction from the Post Correspondence Problem, PCP). - (B) Membership problem for CFG (w ∈ L(G)): Decidable (e.g., using the CYK algorithm in O(n3) time). - (C) Emptiness problem for CFG (L(G) = ∅): Decidable (by finding reachable and generating non-terminals). - (D) Finiteness problem for CFG (|L(G)| < ∞): Decidable (by checking for cycles involving useful non-terminals in the reduced grammar/derivation graph). Therefore, the undecidable problem is (A).

B) Deciding if a given string is generated by a given context-free grammar.

C) Deciding if the language generated by a given context-free grammar is empty.

D) Deciding if the language generated by a given context-free grammar is finite.

Turing Machines and Undecidability

GATE CSE 2014 Set 3 · Q35

MCQ2 marksEasy

Time00:00

Which one of the following problems is undecidable?

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Official answer: A) Deciding if a given context-free grammar is ambiguous.

Turing Machines and Undecidability

Let us analyze the decision problems for Context-Free Grammars (CFGs) / Context-Free Languages (CFLs):

(A) Ambiguity of a CFG: Deciding whether an arbitrary CFG is ambiguous is undecidable (proven via reduction from the Post Correspondence Problem, PCP).
(B) Membership problem for CFG ( $w \in L(G)$ ):Decidable(e.g., using the CYK algorithm in $O(n^3)$ time).
(C) Emptiness problem for CFG ( $L(G) = \emptyset$ ):Decidable (by finding reachable and generating non-terminals).
(D) Finiteness problem for CFG ( $|L(G)| < \infty$ ):Decidable (by checking for cycles involving useful non-terminals in the reduced grammar/derivation graph).

Therefore, the undecidable problem is (A).

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This Turing Machines and Undecidability previous year question is Q35 of the GATE CSE 2014 Set 3 paper. Find more GATE CSE PYQs on Turing Machines and Undecidability in Theory of Computation, or browse all GATE CSE PYQs.

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