1
GATE CSE 2014 Set 3
Numerical
+1
-0
The length of the shortest string NOT in the language (over $$\sum { = \left\{ {a,\,\,b} \right\}}$$) of the following regular expression is ____________. $$a{}^ * b{}^ * \left( {ba} \right){}^ * a{}^ *$$\$
2
GATE CSE 2014 Set 3
+2
-0.6
Consider the following languages over the alphabet $$\sum { = \left\{ {0,\,1,\,c} \right\}:}$$
\eqalign{ & {L_1} = \left\{ {{0^n}\,{1^n}\,\left| {n \ge } \right.0} \right\} \cr & {L_2} = \left\{ {wc{w^r}\,\left| {w \in \left\{ {0,\,1} \right\}{}^ * } \right.} \right\} \cr & {L_3} = \left\{ {w{w^r}\,\left| {w \in \left\{ {0,\,1} \right\}{}^ * } \right.} \right\} \cr}

Here, $${w^r}$$ is the reverse of the string $$w.$$ Which of these languages are deterministic Context- free languages?

A
None of the languages
B
$$(B)$$ Only $${L_1}$$
C
Only $${L_1}$$ and $${L_2}$$
D
All the three languages.
3
GATE CSE 2014 Set 3
+1
-0.3
Let $$\sum \,$$ be a finite non - empty alphabet and let $${2^{\sum {{}^ * } }}$$ be the power set of $$\sum {{}^ * .\,}$$ Which one of the following is TRUE?
A
Both $${2^{\sum {{}^ * } }}$$ and $$\sum {^ * }$$ are countable
B
$${2^{\sum {{}^ * } }}$$ is countable and $$\sum {^ * }$$ is uncountable
C
$${2^{\sum {{}^ * } }}$$ is uncountable and $$\sum {^ * }$$ is countable
D
Both $${2^{\sum {{}^ * } }}$$ and $$\sum {^ * }$$ are uncountable
4
GATE CSE 2014 Set 3
+2
-0.6
Which one of the following problems is un-decidable?
A
Deciding if a given context-free grammar is ambiguous.
B
Deciding if a given string is generated by a given context-free grammar.
C
Deciding if the language generated by a given context-free grammar is empty.
D
Deciding if the language generated by a given context-free grammar is finite.
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