GATE CSE 2007 | Question: 6

Question

Which of the following problems is undecidable?

• A. Membership problem for CFGs
• B. Ambiguity problem for CFGs
• C. Finiteness problem for FSAs
• D. Equivalence problem for FSAs

Comments

Membership problem for CFGs, we have well known CYK algorithms.

• https://web.cs.ucdavis.edu/~rogaway/classes/120/winter12/CYK.pdf
• http://www.cs.nuim.ie/~jpower/Courses/Previous/parsing/node44.html

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Find Video Solution Below:

Video Solution

Answer 1

• Membership problem is decidable as it can be solved by parsers.
• Finiteness problem is decidable for FSAs (also for CFGs), as we just need to check for a loop in the DFA.
• Equivalence problem for FSAs is decidable as we can take the complement of one FSA (complement of FSA is another FSA), and do an intersection with the other (FSAs are closed under intersection also), giving a new FSA. If this new FSA accept no string, then the given FSAs are equivalent, else not equivalent. 
• Only ambiguity problem for CFGs are undecidable.

http://gatecse.in/wiki/Grammar:_Decidable_and_Undecidable_Problems

Correct Answer: $B$

Comments

The equivalence of an FSA needs to be checked both ways,  right?

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Updated link for the link given in answer:

https://gatecse.in/grammar-decidable-and-undecidable-problems/

Answer 2

equivalence of Regular languages is decidable.

1.Membership,

2.Emptiness,

3.Finiteness,

4.Equivalence,

5.Ambiguity,

6.Regularity,

7.Everything,

8.Disjointedness...

All are decidable for Regular languages.

First 3 for CFL.

Only 1^st for CSL and REC.

None for RE.

So option B will be correct one.