Undecidability

Question 11

Consider the following statements:

1. The complement of every Turning decidable 
   language is Turning decidable
2. There exists some language which is in NP 
   but is not Turing decidable
3. If L is a language in NP, L is Turing decidable 

Which of the above statements is/are True?

• Only 2
• Only 3
• Only 1 and 2
• Only 1 and 3

Question 12

Let L(R) be the language represented by regular expression R. Let L(G) be the language generated by a context free grammar G. Let L(M) be the language accepted by a Turing machine M. Which of the following decision problems are decidable? I. Whether L(G) is deterministic context free language? II. Whether L(G1)∩L(G2) is a context free language, where G1 and G2 are deterministic grammar? III. Given a context-free grammar G, is L(G)=Σ^∗ for some alphabet Σ? IV. Given a Turing machine M and a string w, is w ∈ L(M)?

• III and IV only
• II and IV only
• I and II only
• None of the above

Question 13

Which of the following decision problems are undecidable

• I and IV only
• II and III only
• III and IV only
• II and IV only

Question 14

Given ∑ = {a, b}, which of the following sets is not countable ?

• Set of all strings over ∑
• Set of all languages over ∑
• Set of all regular languages over ∑
• Set of all languages over ∑ accepted by Turing machines

Question 15

Which one of the following is not decidable?

• Given a Turing machine M, a string s and an integer k, M accepts s within k steps
• Equivalence of two given Turing machines
• Language accepted by a given finite state machine is not empty
• Language generated by a context free grammar is non-empty

Question 16

Which of the following statements is false?

• The Halting Problem of Turing machines is undecidable
• Determining whether a context-free grammar is ambiguous is undecidable
• Given two arbitrary context-free grammars G1 and G2 it is undecidable whether L(G1)=L(G2)
• Given two regular grammars G1 and G2 it is undecidable whether L(G1)=L(G2)

Question 17

Which of the following statement is false?

• Checking the ambiguity of CFL is decidable.
• Checking whether a given context free language is regular is decidable.
• Checking whether a given context free language is empty is decidable.
• Both A and B

Question 18

Consider the following problems. L(G) denotes the language generated by a grammar G. L(M) denotes the language accepted by a machine M. (I) For an unrestricted grammar G and a string w, whether wϵL(G) (II) Given a Turing machine M, whether L(M) is regular (III) Given two grammar G₁ and G₂, whether L(G₁) = L(G₂) (IV) Given an NFA N, whether there is a deterministic PDA P such that N and P accept the same language Which one of the following statement is correct?

• Only I and II are undecidable
• Only II is undecidable
• Only II and IV are undecidable
• Only I, II and III are undecidable

Question 19

Let S be an NP-complete problem. Q and R are other two problems not known to be NP. Q is polynomial time reducible to S and S is polynomial time reducible to R. Which of the following statements is true?

• R is NP-complete
• R is NP-hard
• Q is NP-complete
• Q is NP-hard

Question 20

Which of the following problems is undecidable?

• To determine if two finite automata are equivalent
• Membership problem for context free grammar
• Finiteness problem for finite automata
• Ambiguity problem for context free grammar