Context free grammar problem for number of a's is twice the number of b's

Question

Can you to find for me a context free grammar for the following language?

$$\{w\in\{a,b\}^*: \#_a(w)=2\#_b(w)\} $$

Here $\#_a(w)$ counts the number of $a$'s in $w$.

Answer 1

Let $c(w)=2\#_b(w)-\#_a(w)$. Your language is exactly $\{w\mid c(w)=0\}$.

Now consider a string $w$ with $c(w)=0$ and $|w|\ge 2$. If we can split $w$ into three parts: $w=pms$ such that $c(p)$ and $c(pm)$ have different signs, i.e., $c(p)c(pm)<0$, then there are two cases:

1. $c(p)>0$ and $c(pm)<0$. Note each time we append a character to string $x$, $c(x)$ is either increased by 2 or decreased by 1, so we can always split $m$ into two parts $m_1m_2$ such that $c(pm_1)=0$.

2. $c(p)<0$ and $c(pm)>0$. In this case we can always split $m$ into two parts $m_1m_2$ such that either $c(pm_1)=0$, or $c(pm_1)=-1$ and $m_2$ begins with $b$.

As a result, either $w=w_1w_2$ where $c(w_1)=c(w_2)=0$, or $w=w_1bw_2$ where $c(w_1)=c(w_2)=-1$.

If $w$ cannot be split into such three parts, then $w$ must be either $bw_1aa$ or $aaw_1b$ where $c(w_1)=0$.

So, if we let $S_-$ generate the language of strings where $2\#_b-\#_a=-1$, then we have the following grammar \begin{align} S&\rightarrow SS\mid S_-bS_-\mid bSaa\mid aaSb\mid\epsilon\\ S_-&\rightarrow aS\mid SS_- \end{align}

Answer 2

The following grammar generates the language $L = \{ w \in \{a, b\}^*\ | \ \#_a(w) = 2\#_b(w) \}$.

S -> bSaa | baSa |
     aSba | abSa |
     aSab | aaSb |
     SS | ε

First, for every b introduced, 2 a's are introduced. The location of an b could be before, between, or after both a's.

Aside for the rule S -> SS, the grammar generates the string from outside in. Thus, there is no left or right recursion. So there are two places in each rule where recursion could happen: inside on the left and inside on the right.

Suppose there is a string $w = bxb \in L$. An example of such a string would be baaaab. Since $\#_a(w) = 2\#_b(w)$, then $\#_a(x) = 2(\#_b(x) + 2)$. There must exists at least one point in $x$ such that $x = x_1x_2$ where $bx_1 \in L$ and $x_2b \in L$. Hence the rule S -> SS.

Edit: The following rules must be added. See the accepted answer for a complete solution. These rules allows us to build string from inside out.

S -> QbQ
Q -> aS | SQ