1
GATE CSE 2016 Set 2
MCQ (Single Correct Answer)
+1
-0.3
Consider the following types of languages: $${L_1}:$$ Regular, $${L_2}:$$ Context-free, $${L_3}:$$ Recursive, $${L_4}:$$ Recursively enumerable. Which of the following is/are TRUE?

$$\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,$$ $$\overline {{L_3}} \cup {L_4}$$ is recursively enumerable
$$\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,\,\,$$ $$\overline {{L_2}} \cup {L_3}$$ is recursive
$$\,\,\,{\rm I}{\rm I}{\rm I}.\,\,\,\,\,\,\,$$ $$L_1^ * \cap {L_2}$$ is context-free
$$\,\,\,{\rm I}V.\,\,\,\,\,\,\,$$ $${L_1} \cup \overline {{L_2}} $$ is context-free

A
$${\rm I}$$ only
B
$${\rm I}$$ and $${\rm III}$$ only
C
$${\rm I}$$ and $${\rm IV}$$ only
D
$${\rm I},$$ $${\rm II}$$ and $${\rm III}$$ only
2
GATE CSE 2015 Set 2
MCQ (Single Correct Answer)
+1
-0.3
Consider the following statements.

$$\,\,\,$$ $${\rm I}.\,\,\,\,\,\,\,\,\,$$ The complement of every Turing decidable language is Turing decidable
$$\,$$ $${\rm II}.\,\,\,\,\,\,\,\,\,$$ There exists some language which is in $$NP$$ but is not Turing decidable
$${\rm III}.\,\,\,\,\,\,\,\,\,$$ If $$L$$ is a language in $$NP,$$ $$L$$ is Turing decidable

Which of the above statements is/are true?

A
Only $${\rm I}$$$${\rm I}$$
B
Only $${\rm III}$$
C
Only $${\rm I}$$ and $${\rm II}$$
D
Only $${\rm I}$$ and $${\rm III}$$
3
GATE CSE 2015 Set 1
MCQ (Single Correct Answer)
+1
-0.3

For any two languages L1 and L2 such that L1 is context-free and L2 is recursively enumerable but not recursive, which of the following is/are necessarily true?

I. $${\overline L _1}$$ (complement of L1) is recursive
II. $${\overline L _2}$$ (complement of L2) is recursive
III. $${\overline L _1}$$ is context-free
IV. $${\overline L _1} \cup {L_2}$$ is recursively enumerable
A
I only
B
III only
C
III and IV only
D
I and IV only
4
GATE CSE 2014 Set 2
MCQ (Single Correct Answer)
+1
-0.3
Let $$A\,\,{ \le _m}\,\,B$$ denotes that language $$A$$ is mapping reducible (also known as many-to-one reducible) to language $$B.$$ Which one of the following is FALSE?
A
If $$A\,\,{ \le _m}\,\,B$$ and $$B$$ is recursive then $$A$$ is recursive.
B
If $$A\,\,{ \le _m}\,\,B$$ and $$A$$ is undecidable then $$B$$ is un-decidable.
C
If $$A\,\,{ \le _m}\,\,B$$ and $$B$$ is recursively enumerable then $$A$$ is recursively enumerable.
D
If $$A\,\,{ \le _m}\,\,B$$ and $$B$$ is not recursively enumerable then $$A$$ is not recursively enumerable.
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