1
GATE CSE 2006
+2
-0.6
Let $${L_1}$$ be a regular language, $${L_2}$$ be a deterministic context-free language and $${L_3}$$ a recursively enumerable, but not recursive, language. Which one of the following statement is false?
A
$${L_1} \cap {L_2}$$ is deterministic $$CFL$$
B
$${L_3} \cap {L_1}$$ is recursive
C
$${L_1} \cup {L_2}$$ is context-free
D
$${L_1} \cap {L_2} \cap {L_3}$$ is recursively enumerable
2
GATE CSE 2005
+2
-0.6
Let $${L_1}$$ be a recursive language, and Let $${L_2}$$ be a recursively enumerable but not a recursive language. Which one of the following is TRUE?
A
$$\overline {{L_1}}$$ is recursive and $$\overline {{L_2}}$$ is recursively enumerable
B
$$\overline {{L_1}}$$ is recursive and $$\overline {{L_2}}$$ is not recursively enumerable
C
$$\overline {{L_1}}$$ and $$\overline {{L_2}}$$ are recursively enumerable
D
$$\overline {{L_1}}$$ is recursively enumerable and $$\overline {{L_2}}$$ is recursive
3
GATE CSE 2003
+2
-0.6
A single tape Turing Machine $$M$$ has two states $${q_0}$$ and $${q_1}$$, of which $${q_0}$$ is the starting state. The tape alphabet of $$M$$ is $$\left\{ {0,\,\,1,\,\,B} \right\}$$ and its input alphabet is $$\left\{ {0,\,\,1} \right\}$$. The symbol $$B$$ is the blank symbol used to indicate end of an input string. The transition function of $$M$$ is described in the following table. The table is interpreted as illustrated below. The entry $$\left( {{q_1},1,\,R} \right)$$ in row $${{q_0}}$$ and column $$1$$ signifies that if $$M$$ is in state $${{q_0}}$$ and reads $$1$$ on the current tape square, then it writes $$1$$ on the same tape square, moves its tape head one position to the right and transitions to state $${{q_1}}$$.

Which of the following statements is true about $$M?$$

A
$$M$$ does not halt on any string in $${\left( {0 + 1} \right)^ + }$$
B
$$M$$ does not halt on any string in $${\left( {00 + 1} \right)^ + }$$
C
$$M$$ halts on all string ending in a $$0$$
D
$$M$$ halts on all string ending in $$a$$
4
GATE CSE 2003
+2
-0.6
Define Languages $${L_0}$$ and $${L_1}$$ as follows
$${L_0} = \left\{ { < M,\,w,\,0 > \left| {M\,\,} \right.} \right.$$ halts on $$\left. w \right\}$$
$${L_1} = \left\{ { < M,w,1 > \left| M \right.} \right.$$ does not halts on $$\left. w \right\}$$

Here $$< M,\,w,\,i >$$ is a triplet, whose first component, $$M$$ is an encoding of a Turing Machine, second component, $$w$$, is a string, and third component, $$t,$$ is a bit.
Let $$L = {L_0} \cup {L_1}.$$ Which of the following is true?

A
$$L$$ is recursively enumerable, but $$\overline L$$ is not
B
$$\overline L$$ is recursively enumerable, but $$L$$ is not
C
Both $$L$$ and $$\overline L$$ are recursive
D
Neither $$L$$ nor $$\overline L$$ is recursive enumerable
GATE CSE Subjects
Theory of Computation
Operating Systems
Algorithms
Digital Logic
Database Management System
Data Structures
Computer Networks
Software Engineering
Compiler Design
Web Technologies
General Aptitude
Discrete Mathematics
Programming Languages
Computer Organization
EXAM MAP
Joint Entrance Examination