How to convert regular expression to CFG?

Question

How can I convert the regular expression (ab*)*b to a context-free grammar?

When I look for examples I keep seeing plus signs in the expression but I don’t have any. Is that just a different way of writing it?

Answer 1

There is a simple algorithm to convert regular expressions to context-free grammars. It goes as follows.

Base cases:

• $\emptyset$ corresponds to the empty grammar.
• $\epsilon$ corresponds to the grammar $S \to \epsilon$.
• $\sigma$ (where $\sigma \in \Sigma$) corresponds to the grammar $S \to \sigma$.

Inductive cases:

• $r = r_1 + r_2$. Given CFGs for $r_1,r_2$ with disjoint nonterminals and starting symbols $S_1,S_2$, add rules $S \to S_1 \mid S_2$ and make $S$ the new starting symbols.
• $r = r_1r_2$. Same, adding the rule $S \to S_1 S_2$ instead.
• $r = r_1{}^*$. Same, adding the rules $S \to S S_1 \mid \epsilon$ instead.