GATE|| TOC (Undecideability) ||Pyq (2010 to2025)

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Question 1

Consider the following sets:
S1. Set of all recursively enumerable languages over the alphabet {0,1}
S2. Set of all syntactically valid C programs
S3. Set of all languages over the alphabet {0,1}
S4. Set of all non-regular languages over the alphabet {0,1}
Which of the above sets are uncountable?

[GATE 2019|| MCQ|| 2-mark]

  • S2 and S3

  • S3 and S4

  • S1 and S4

  • S1 and S2

Question 2

Which of the following statements is false?

[GATE || CS || PYQ || 2008]

  • Every NFA can be converted to an equivalent DFA

  • Every non-deterministic Turing machine can be converted to an equivalent deterministic Turing machine

  • Every regular language is also a context-free language

  • Every subset of a recursively enumerable set is recursive

Question 3

Let L1 be a recursive language, and let L2 be a recursively enumerable but not a recursive language. Which one of the following is TRUE?


[GATE || CS || PYQ || 2005]

  • L1' is recursive and L2' is recursively enumerable

  • L1' is recursive and L2' is not recursively enumerable

  • L1' and L2' is recursively enumerable

  • L1' is recursively enumerable and L2' is recursive

Question 4

Let L= L1 ∩ L2 , where L1 and L2 are languages as defined below:

L1={ambmcanbn | m,n >=0}

L2={aibjck | i,j,k >=0}

Then is L

[GATE || CS || PYQ || 2009]

  • Not recursive

  • Regular

  • Context free but not regular

  • Recursively enumerable but not context free

Question 5

Which of the following is true for the language

{ ap| p is prime} ?

[GATE || CS || PYQ || 2008]

  • It is not accepted by a Turing Machine

  • It is regular but not context-free

  • It is context-free but not regular

  • It is neither regular nor context-free, but accepted by a Turing machine

Question 6

The language L={ oi21i ∣ i≥0 } over the alphabet {0,1,2} is:

[GATE || CS || PYQ || 2007]

  • not recursive

  • is recursive and is a deterministic CFL

  • is a regular language

  • is not a deterministic CFL but a CFL

Question 7

Let L1 be a regular language, L2 be a deterministic context-free language and L3 a recursively enumerable, but not recursive, language. Which one of the following statements is false?

[GATE || CS || PYQ || 2006]

  • L1 ∩ L2 is a deterministic CFL

  • L3 ∩ L1 is recursive

  • L1 U L2 is context free

  • L1 ∩ L2 ∩ L3 is recursively enumerable

Question 8

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

[GATE || CS || PYQ || 2008]

  • I and II

  • I and IV

  • II and III

  • II and IV

Question 9

Which of the following problems is undecidable?

[GATE || CS || PYQ || 2007]

  • Membership problem for CFGs

  • Ambiguity problem for CFGs

  • Finiteness problem for FSAs

  • Equivalence problem for FSAs

Question 10

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?

[GATE || CS || PYQ || 2005 ]

  • 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

There are 28 questions to complete.

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