Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to write a context-free grammar for this language:

$L = \{a^n b a^m (bb)^n : m \ge 1, n \ge 0\}$

I was getting lost with maintaining $n$ number of $a$'s and $(bb)$'s and I'm not sure how to fix it. The terminals don't seem to be working out because I think splitting it up between lines is just asking for the $n$ to not be the same between the 2 terminals that require it (as I stated above). The $m$ seems ok though:

$S \rightarrow a S b b \mid A$

$A \rightarrow b B$

$B \rightarrow a B \mid a$

Would be very helpful if someone can get me back on track.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

It looks fine: there’s nothing there to be fixed. Any derivation must look like this:

$$S\Rightarrow^n a^nSb^{2n}\Rightarrow a^nb^{2n}\Rightarrow a^nbBb^{2n}\Rightarrow^m a^nba^mBb^{2n}\Rightarrow a^nba^{m+1}b^{2n}\;,$$

where $n$ is the number of times you use $S\to aSbb$, and $m$ is the number of times you use $B\to aB$. Since $n,m\ge 0$, this gives you exactly what you want.

share|improve this answer
    
Great thanks, it didn't feel like I was including everything but yes you showed me how I was. +1 and accepted. –  stackuser Nov 3 '13 at 15:52
    
@stackuser: You’re welcome. (What I did there is a useful technique if your grammar doesn’t do much branching.) –  Brian M. Scott Nov 3 '13 at 16:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.