1
GATE CSE 2002
+2
-0.6
The Finite state machine described by the following state diagram with $$A$$ as starting state, where an arc label is $$x/y$$ and $$x$$ stands for $$1-bit$$ input and $$y$$ stands for $$2$$-bit output
A
Outputs the sum of the present and the previous bits of the input.
B
Outputs $$01$$ whenever the input sequence contains $$11$$
C
Outputs $$00$$ whenever the input sequence contains $$10$$
D
None of the above
2
GATE CSE 2001
+2
-0.6
Consider a $$DFA$$ over $$\sum { = \left\{ {a,\,\,b} \right\}}$$ accepting all strings which have number of $$a'$$s divisible by $$6$$ and number of $$b'$$s divisible by $$8$$. What is the minimum number of states that the $$DFA$$ will have?
A
$$8$$
B
$$14$$
C
$$15$$
D
$$48$$
3
GATE CSE 2001
+2
-0.6
Consider the following languages:
$${L_1} = \left\{ {w\,w\left| {w \in {{\left\{ {a,\,b} \right\}}^ * }} \right.} \right\}$$
$${L_2} = \left\{ {w\,{w^R}\left| {w \in {{\left\{ {a,\,b} \right\}}^ * },} \right.{w^R}\,\,} \right.$$ is the reverse of $$\left. w \right\}$$
$${L_3} = \left\{ {{0^{2i}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$
$${L_4} = \left\{ {{0^{{i^2}}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$

Which of the languages are regular?

A
Only $${L_1}$$ and $${L_2}$$
B
Only $${L_2},$$ $${L_3}$$ and $${L_4}$$
C
Only $${L_3}$$ and $${L_4}$$
D
Only $${L_3}$$
4
GATE CSE 2000
+2
-0.6
What can be said about a regular language $$L$$ over $$\left\{ a \right\}$$ whose minimal finite state automation has two states?
A
Must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is odd $$\left. \, \right\}$$
B
Must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is even $$\left. \, \right\}$$
C
Must be $$\left\{ {{a^n}\left| {n \ge } \right.\,\,} \right.0\left. \, \right\}$$
D
Either $$L$$ must be $$\left\{ {{a^n}\left| n \right.\,\,} \right.$$ is odd$$\left. \, \right\}\,\,$$ or $$L$$ must be $$\left\{ {{a^n}\left| n \right.} \right.$$ is even$$\left. \, \right\}$$
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