# Undecidability

Question 1

```Which of the following problems are decidable?
```
• 1, 2, 3, 4
• 1, 2
• 2, 3, 4
• 3, 4

Question 2

Which of the following is/are undecidable?
• 3 only
• 3 and 4 only
• 1, 2 and 3 only
• 2 and 3 only

Question 3

Which of the following are decidable?
```I. Whether the intersection of two regular languages is infinite
II. Whether a given context-free language is regular
III. Whether two push-down automata accept the same language
IV. Whether a given grammar is context-free```
• I and II
• I and IV
• II and III
• II and IV

Question 4

Which of the following problems is undecidable? [2007]
• Membership problem for CFGs
• Ambiguity problem for CFGs.
• Finiteness problem for FSAs.
• Equivalence problem for FSAs.

Question 5

Let be the encoding of a Turing machine as a string over ∑= {0, 1}.  Let L = { |M is a Turing machine that accepts a string of length 2014 }. Then, L is
• decidable and recursively enumerable
• undecidable but recursively enumerable
• undecidable and not recursively enumerable
• decidable but not recursively enumerable

Question 6

Which of the following problems is undecidable?
• Deciding if a given context-free grammar is ambiguous.
• Deciding if a given string is generated by a given context-free grammar.
• Deciding if the language generated by a given context-free grammar is empty.
• Deciding if the language generated by a given context-free grammar is finite.

Question 7

Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?
• P3 is decidable if P1 is reducible to P3
• P3 is undecidable if P3 is reducible to P2
• P3 is undecidable if P2 is reducible to P3
• P3 is decidable if P3 is reducible to P2\'s complement

Question 8

Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 if f(x) ∈ L2]. Further, let f-1 be also polynomial time computable. Which of the following CANNOT be true?

• L1 ∈ P and L2 is finite

• L1 ∈ NP and L2 ∈ P

• L1 is undecidable and L2 is decidable

• L1 is recursively enumerable and L2 is recursive

Question 9

Consider the following problem X.
```Given a Turing machine M over the input alphabet Σ, any
state q of M And a word w∈Σ*, does the computation of M
on w visit the state q? ```
Which of the following statements about X is correct?
• X is decidable
• X is undecidable but partially decidable
• X is undecidable and not even partially decidable
• X is not a decision problem

Question 10

Consider the following decision problems:

```(P1) Does a given finite state machine accept a given string
(P2) Does a given context free grammar generate an infinite
number of strings```

Which of the following statements is true?

• Both (P1) and (P2) are decidable

• Neither (P1) nor (P2) are decidable

• Only (P1) is decidable

• Only (P2) is decidable

There are 27 questions to complete.

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