1
GATE CSE 2006
+2
-0.6
If $$s$$ is a string over $${\left( {0 + 1} \right)^ * }$$ then let $${n_0}\left( s \right)$$ denote the number of $$0'$$ s in $$s$$ and $${n_1}\left( s \right)$$ the number of $$1'$$s in $$s.$$ Which one of the following languages is not regular?
A
$$L = \left\{ {s \in {{\left( {0 + 1} \right)}^ * }\left| {{n_0}\left( s \right)\,\,} \right.} \right.$$ is a $$3$$-digit prime$$\left. \, \right\}$$
B
$$L = \left\{ {s \in {{\left( {0 + 1} \right)}^ * }\left| {\,\,} \right.} \right.$$ for every prefix $$s'$$ of $$s.$$ $$\,\left| {{n_0}\left( {{s^,}} \right) - {n_1}\left( {{s^,}} \right)\left| { \le \left. 2 \right\}} \right.} \right.$$
C
$$L = \left\{ {s \in {{\left( {0 + 1} \right)}^*}\left\| {{n_0}\left( s \right) - {n_1}\left( s \right)\left| { \le \left. 4 \right\}} \right.} \right.} \right.$$
D
$$L = \left\{ {s \in {{\left( {0 + 1} \right)}^ * }} \right.\left| {{n_0}\left( s \right)} \right.$$ mod $$7 = {n_1}\left( s \right)$$ mod $$5 = \left. 0 \right\}$$
2
GATE CSE 2006
+2
-0.6
Consider the regular language $$L = {\left( {111 + 11111} \right)^ * }.$$ The minimum number of states in any $$DFA$$ accepting this language is
A
$$3$$
B
$$5$$
C
$$8$$
D
$$9$$
3
GATE CSE 2005
+2
-0.6
Consider the machine $$M:$$ The language recognized by $$M$$ is:
A
$$\left\{ {w\,\, \in \,\left\{ {a,b} \right\}{}^ * \left| \, \right.} \right.$$ every $$a$$ in $$w$$ is followed by exactly two $$\left. {b's} \right\}$$
B
$$\left\{ {w\,\, \in \,\left\{ {a,b} \right\}{}^ * \left| \, \right.} \right.$$ every $$a$$ in $$w$$ is followed by at least two $$\left. {b's} \right\}$$
C
$$\left\{ {w\,\, \in \,\left\{ {a,b} \right\}{}^ * \left| \, \right.} \right.$$ $$w$$ contains the substring $$\,\left. {'abb'\,\,} \right\}\,$$
D
$$\left\{ {w\,\, \in \,\left\{ {a,b} \right\}{}^ * \left| \, \right.} \right.$$ $$w$$ does not contain $$'aa'$$ as a substring$$\left. \, \right\}$$
4
GATE CSE 2004
+2
-0.6
The following finite state machine accepts all those binary strings in which the number of $$1's$$ and $$0's$$ are respectively
A
Divisible by $$3$$ and $$2$$
B
Odd and even
C
Even and odd
D
Divisible by $$2$$ and $$3$$
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