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Consider the following two statements:

$S_1: \left\{ 0^{2n} \mid n \geq 1 \right\}$ is a regular language

$S_2: \left\{0^m1^n0^{m+n} \mid m \geq 1 \text{ and } n \geq 1 \right\}$ is a regular language

Which of the following statement is correct?

  1. Only $S_1$ is correct
  2. Only $S_2$ is correct
  3. Both $S_1$ and $S_2$ are correct
  4. None of $S_1$ and $S_2$ is correct
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2 Answers

Best answer
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25 votes

Only $S_1$ is correct!

A DFA with $3$ states will be needed, as the strings in the language $S_1$ are $00, 0000, 000000,$ and so on. which is the set of all strings with even number of $0's$ and with length greater than $0$. We would have needed only $2$ sates had empty string also been in the language but $n\geq 1$ prohibits it and so we need $3$ states in our DFA. This assumes that the language is over $\{0\}$ and not $\{0,1\}.$

$S_2$ is DCFL as we need to do infinite counting of $0's$ and $1's$ here.

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15 votes
15 votes

DFA For S1 :

S2 is PDA since stack is needed for comparision finite memory is not sufficient

              Till 1 push in stack after that pop from stack for every 0.

Answer:

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