Convert regex for comma separated value to CFG grammar

Question

I have the following regex that I'm trying to convert to a CFG: e|a(,a)* (e representing the empty state). Basically I want to match a comma-separated list (without a leading or trailing comma) or nothing at all. Here are some of my attempts:

S = A
A = e | a | a , A

This doesn't work, it matches ,a (leading comma).

S = A | B
A = e
B = a C
C = , C | a | e

This doesn't work either, it matches a, and aa (trailing and missing comma).

Is my regex even representable as a CFG?

Thanks in advance

Answer 1

At the heart of your regex is an alternation {$\alpha|\beta$), which means that the regex matches $\alpha$ or $\beta$. That's how you should write the CFG: $$\begin{align}S\to&\;\alpha\\ |&\;\beta \end{align}$$ Here, $\alpha$ is $\varepsilon$ and $\beta$ is the subexpression $a(,a)^*$, for which we'll define the non-terminal $A$.

It's easy to see what $A$ is: it starts with an $a$ and can be extended arbitrarily with $, a$: $$\begin{align}S\to&\;\varepsilon\\ |&\;A\\ A\to&\;a\\ |&\;A\;,a \end{align}$$