GATE CS 2017 Set 2 — Question 41
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Theory of Computation → Context-Free Languages → Decision Problems (CFL)
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Question
Let
II. Given a context-free grammar , is ?
III. Given a context-free grammar , is for some alphabet ?
IV. Given a Turing machine and a string , is ?
L(R) be the language represented by regular expression . Let L(G) be the language generated by a context free grammar . Let L(M) be the language accepted by a Turing machine . Which of the following decision problems are undecidable?I. Given a regular expression and a string , is ?II. Given a context-free grammar , is ?
III. Given a context-free grammar , is for some alphabet ?
IV. Given a Turing machine and a string , is ?
Correct answer
(D) III and IV only
Solution
Analysis of decision problems:
- I. Membership for Regular Languages: Decidable. We can construct a DFA from and check if it accepts .
- II. Emptiness for CFG: Decidable. There are algorithms to check if a CFG generates any terminal string.
- III. Totality for CFG: Undecidable. Checking if a CFG generates all possible strings over its alphabet is a known undecidable problem.
- IV. Membership for Turing Machines: Undecidable. This is equivalent to the Halting Problem (specifically, the Acceptance Problem ), which is undecidable.
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