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Given the language:

$$ L=\{ \omega \in \{ a,b,c,d \}^* : |\omega|_a = |\omega|_b \} $$

I propose the following grammar: $$ \begin{align*} S &\to \varepsilon \mid aSbS \mid bSaS \\ S &\to cS \mid dS \mid Sc \mid Sd \end{align*} $$

I want to know if this is correct or if there exists a smarter grammar (a shorter grammar).

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I'm going to address the second part of the question first, about a "smarter grammar." One improvement that can be made is omitting the rules

$$S \rightarrow Sc \hspace{10px}| \hspace{10px} Sd.$$

Assuming we have made that change, to answer the first part, if the grammar is correct we need to answer two questions.

  1. Does $S$ only generate strings within $L$?
  2. Can every string in $L$ be generated by $S$?

If the answer to these questions is both yes, then a string is in $L$ if and only if it is generated by $S$, which means that $S$ is a correct grammar for the language.

The answer to 1. is clearly yes because every string generated by $S$ must have the same number of $a$'s as it does $b$'s because in every rule where an $a$ is present a $b$ is also present, and vise versa.

The answer to number 2 is a bit trickier. We will use induction on $|\omega|_a = |\omega|_b$ to show that any string in $L$ can be generated by $S$

(Base case $n=0$) If $|\omega|_a = |\omega|_b = 0$, then $\omega$ consists of a sequence of $c$'s and $d$'s, and can be generated by scanning over it using the rules

$$S \rightarrow cS \hspace{10px} | \hspace{10px} dS$$

until we terminate with $S \rightarrow \epsilon$.

($n$ implies $n+1$) Assume by induction that for all $\omega \in L$ with $|\omega|_a = |\omega|_b \leq n$ that $\omega$ can be generated by $S$. Now let $\omega' \in L$ be such that $|\omega'|_a = |\omega'|_b = n+1$.

At some point in $\omega'$ there must be at least one time that an $a$ comes before a $b$ (with only $c$'s and $d$'s between them) or a $b$ comes before an $a$ (with only $c$'s and $d$'s between them). Without loss of generality, we can assume there is an $a$ that comes before a $b$.

Now, we can decompose $\omega' = XaYbZ$ where $|X|_a + |Z|_a = |XZ|_a = n$ (and the same for $b$) and $Y \in \{c,d\}^*$ . This holds because we are taking one $a$ and one $b$ from $\omega'$ which had $n+1$ of each which leaves $n$ in the rest of the string, and we know that there is a pair of $a$ and $b$ with just $c$ and $d$ between.

Now we can get a little clever and observe that the string $XcZ \in L$ and $|XcZ|_a = |XcZ|_b = n$ and therefore, by the inductive hypothesis, $XcZ$ can be generated by $S$.

Now here's the tricky part. Consider the process that generated $XcZ$. At some point in the generation, we must have used the rule $S \rightarrow cS$ to generate the $c$ between $X$ and $Z$. If we instead chose to use

$$S \rightarrow aS^*bS$$

as the expansion where $S^*$ generates $Y$ as in the case where $n = 0$, and did everything else exactly the same, then what happens is we replace the $c$ in $XcZ$ with $aYb$, and get the string $XaYbZ$. This demonstrates that $\omega'$ can be generated by $L$ because we have just found a process by which it can be generated by $S$.

Therefore by induction on $|\omega|_a = |\omega|_b$ every string in $L$ can be generated by $S$.

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