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
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