Arjun Suresh (talk | contribs) (→Solution) |
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| + | Let $Σ = \{a, b, c\}$. Which of the following statements is true ? | ||
| − | + | a)For any $A ⊆ Σ^*$, if $A$ is regular, then so is $\{xx | x ∊ A\}$ | |
| − | + | '''b)For any $A ⊆ Σ^*$, if $A$ is regular, then so is $\{x | xx ∊ A\}$''' | |
| − | + | c)For any $A ⊆ Σ^*$, if $A$ is context-free, then so is $\{xx | x ∊ A\}$ | |
| − | + | d)For any $A ⊆ Σ^*$, if $A$ is context-free, then so is $\{x | xx ∊ A\}$ | |
| − | + | ==={{Template:Author|Arjun Suresh|{{arjunweb}} }}=== | |
| + | We can get a DFA for $L = \{x | xx ∊ A\}$ as follows: | ||
| + | Take DFA for $A$ $(Q, \delta, \Sigma, S, F)$ with everything same except initially making $F = \phi$. | ||
| + | Now for each state $D \in Q$, consider 2 separate DFAs, one with $S$ as the start state and $D$ as the final state and another with $D$ as the start state and set of final states $⊆ F$. If both these DFAs accept same language make $D$ as final state. | ||
| − | + | This procedure works as checking the equivalence of 2 DFAs is decidable. | |
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| − | This procedure works as the equivalence of 2 DFAs is decidable. | ||
'''Contradictions for other choices''' | '''Contradictions for other choices''' | ||
| − | a) Consider A = Σ*. Now for $w \in A, L = \{xx | x | + | a) Consider $A = Σ^*$. Now for $w \in A, L = \{xx | x \in A\} = \{ww | w \in Σ^*\} $ which is context sensitive |
c) Same example as for (a) | c) Same example as for (a) | ||
| − | d) $\{a^nb^nc^*a^*b^nc^n|n | + | d)Consider $A = \{a^nb^nc^*a^*b^nc^n|n\ge0\} $ |
This is CFL. But if we make L from A as per (d), it'll be | This is CFL. But if we make L from A as per (d), it'll be | ||
| − | L = {a^nb^nc^n|n | + | $L = \{a^nb^nc^n|n\ge0\}$ which is not context free.. |
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{{Template:FBD}} | {{Template:FBD}} | ||
| − | [[Category:Automata Theory | + | [[Category: Non-GATE Questions from Automata Theory]] |
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Let $Σ = \{a, b, c\}$. Which of the following statements is true ?
a)For any $A ⊆ Σ^*$, if $A$ is regular, then so is $\{xx | x ∊ A\}$
b)For any $A ⊆ Σ^*$, if $A$ is regular, then so is $\{x | xx ∊ A\}$
c)For any $A ⊆ Σ^*$, if $A$ is context-free, then so is $\{xx | x ∊ A\}$
d)For any $A ⊆ Σ^*$, if $A$ is context-free, then so is $\{x | xx ∊ A\}$
We can get a DFA for $L = \{x | xx ∊ A\}$ as follows: Take DFA for $A$ $(Q, \delta, \Sigma, S, F)$ with everything same except initially making $F = \phi$. Now for each state $D \in Q$, consider 2 separate DFAs, one with $S$ as the start state and $D$ as the final state and another with $D$ as the start state and set of final states $⊆ F$. If both these DFAs accept same language make $D$ as final state.
This procedure works as checking the equivalence of 2 DFAs is decidable.
Contradictions for other choices
a) Consider $A = Σ^*$. Now for $w \in A, L = \{xx | x \in A\} = \{ww | w \in Σ^*\} $ which is context sensitive
c) Same example as for (a)
d)Consider $A = \{a^nb^nc^*a^*b^nc^n|n\ge0\} $ This is CFL. But if we make L from A as per (d), it'll be $L = \{a^nb^nc^n|n\ge0\}$ which is not context free..
Let Σ = {a, b, c}. Which of the following statements is true ?
a)For any A ⊆ Σ*, if A is regular, then so is {xx | x ∊ A}
b)For any A ⊆ Σ*, if A is regular, then so is {x | xx ∊ A}
c)For any A ⊆ Σ*, if A is context-free, then so is {xx | x ∊ A}
d)For any A ⊆ Σ*, if A is context-free, then so is {x | xx ∊ A}
We can get a DFA for L as follows: Take DFA for A. Let S be its start state. For each state D of A, consider 2 separate DFAs, one with S as the start state and D as the final state and another with D as the start state and final state in final states of A. If both these DFAs accept same language make D as final state.
This procedure works as the equivalence of 2 DFAs is decidable.
Contradictions for other choices
a) Consider A = Σ*. Now for $w \in A, L = \{xx | x ∊ A\} = \{ww | w \in Σ^*\} $ which is context sensitive
c) Same example as for (a)
d) $\{a^nb^nc^*a^*b^nc^n|n>0\} $ This is CFL. But if we make L from A as per (d), it'll be L = {a^nb^nc^n|n>0} which is not context free..
GATECSE