GATE CS 2016 Set 1 — Question 27
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Theory of Computation → Context-Free Languages → Decision Problems (CFL)
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Question
Which of the following decision problems are undecidable?I. Given NFAs and , is ?
II. Given a CFG and a string , does ?
III. Given CFGs and , is ?
IV. Given a TM , is ?
II. Given a CFG and a string , does ?
III. Given CFGs and , is ?
IV. Given a TM , is ?
Correct answer
(C) III and IV only
Solution
Let's analyze each problem:I. Emptiness of Intersection of Regular Languages: Given NFAs and , their languages and are regular. The intersection of two regular languages is regular. The emptiness problem for regular languages is decidable (reachability in the state graph). Thus, this is decidable.II. Membership Problem for CFGs: Given a CFG and string , determining if is the standard parsing problem. The CYK algorithm can solve this in time. Thus, this is decidable.III. Equivalence Problem for CFGs: Determining if for two context-free grammars is a known undecidable problem. (Note: Equivalence is decidable for DCFGs, but not for general CFGs).IV. Emptiness Problem for Turing Machines: Determining if for a TM is undecidable (Rice's Theorem or reduction from the Halting Problem).Therefore, problems III and IV are undecidable.
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