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Does the following set of production rules produce a regular language or not?

$S \to AB \mid b $
$A \to SB$
$B \to AS \mid a$

I have generated following words with above grammar

  1. $b , baa , baaaa , baaaaaa, .....$
  2. $bbaba , babab , bababaa , bbbabba ,.....$

But I am not able to create a language using it. Is this grammar just context free or is this regular also ? If yes, how to prove that ?

John L.
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S.J.
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    Every regular grammar is context free grammar as well. So, this would be at least $CFG$. Now try if you could convert it into right linear or left linear or not. That will prove whether it's regular as well or not. – Mr. Sigma. Dec 02 '18 at 12:23
  • Here are some further questions. Is this language inherently ambiguous? Can this language be recognized in linear time? – John L. May 02 '19 at 20:37

1 Answers1

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The grammar can be transformed to

$\quad$ $S \to SBB \mid b $
$\quad$ $B \to SBS \mid a$

Let $L_S$ be the language generated by the rules above (with $S$ as the start symbol). $L_S$ is context-free by definition.

We will prove it is not regular. The idea comes from the rule, $B\to SBS$.

Let $L_B$ be the language generated by the two rules above with $B$ as the start symbol.

Claim: We have the following inclusions of languages.
$\quad\{b\}\cup\{b^{i+1}ab^ia\mid i\ge0\}\subseteq L_S$
$\quad\{b^iab^i\mid i\ge0\}\subseteq L_B$

Proof: Use mathematical induction on $n$, the upper bound for $i$.

  • The base case when $n=0$. $S\Rightarrow SBB \Rightarrow^* baa$ and $B\Rightarrow a$.
  • Suppose it is true for $n$. Then $B\Rightarrow SBS\Rightarrow^* b(b^nab^n)b=b^{n+1}ab^{n+1}$. $S\Rightarrow SBB\Rightarrow^* b(b^{n+1}ab^{n+1})a=b^{(n+1)+1}ab^{n+1}a$. So it is true for $n+1$ as well.

Claim: We have following inclusions of languages. $\quad L_S\subseteq E_S :=\{ba^{2i}\mid i\ge0\}\cup\{b^{i+1}ab^ia\mid i>0\}\cup\{\text{words that have at least three } a\text{'s}\}$ $\quad L_B\subseteq E_B:=\{a\}\cup\{\text{words that have at least two } a\text{'s}\}$

Proof: It is easy to check that $E_SE_BE_B\subseteq E_S$, $a\in E_S$, $E_BE_SE_B\subseteq E_B$ and $b\in E_B$.

The above two claims implies that $L_S\cap L(b^*ab^*a) = \{b^{i+1}ab^ia\mid i\ge0\}$, which is not a regular language. Since $L(b^*ab^*a)$ is regular, $L_S$ cannot be regular.

Exercise 1. Show that the following grammar generates a regular language.
$\quad$ $S \to SAS \mid b $
$\quad$ $A \to ASA \mid a$

Exercise 2. Show that the following grammar generates a non-regular language.
$\quad$ $S \to AB \mid b $
$\quad$ $B \to SA $
$\quad$ $A \to BS \mid a$

John L.
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