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+11 votes
734 views

Consider the following grammar G:

$S \rightarrow bS \mid aA \mid b$

$A \rightarrow bA \mid aB$

$B \rightarrow bB \mid aS \mid a$

Let $N_a(w)$ and $N_b(w)$ denote the number of a’s and b’s in a string $\omega$ respectively.

The language $L(G)$ over  $\left\{a, b\right\}^+$ generated by $G$ is

  1. $\left\{w \mid N_a(w) > 3N_b(w)\right\}$

  2. $\left\{w \mid N_b(w) > 3N_a(w)\right\}$

  3. $\left\{w \mid N_a(w) = 3k, k \in \left\{0, 1, 2, …\right\}\right\}$

  4. $\left\{w \mid N_b(w) = 3k, k \in \left\{0, 1, 2, …\right\}\right\}$

asked in Compiler Design by Veteran (69k points) | 734 views
S-> b then S becomes final state.

but how draw transition for B-> a ???
No of a's divisible by 3.

2 Answers

+21 votes
Best answer

above CFG generate string $b$, $aaa$..
$b$ will eliminate options A and D
$aaa$ eliminate options B.
C is answer i.e. number of $a = 3k, k =0,1,2$....

answered by Veteran (53.4k points)
edited by
how we form 0 no's  of a
s-->b , 0 a present
If anyone face it difficult to understand then he/she can draw NFA, as given grammar is right-linear.
how?  not getting    result
+6 votes

becoz it is right linear grammer,,,so draw m/c

so number of a = 3k, k =0,1,2....

answered by Veteran (12.4k points)
edited by
How come is this a DFA?

 Tuhin Dutta haha 

Sir I understand the logic of converting a linear grammar to DFA, but why do we make the productions $S \rightarrow b$ and $B \to a$ lead to a null state? Any reference will be great.

@vaibhav101

In right linear questions,they can be attacked by converting to machine.sometimes we dont recognize language by just looking at it,But when we get machine we can recognize easily.

Now coming to your doubt.That null was because it is a non terminal.If from start symbol we get b,it is end we cannot get anything more.And it is in language so it is accepted state

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