# Find a regular grammar that generates language $L$

I have a language $$L$$ = {$$vabu$$ | $$v$$,$$u\in \{a,b\}^*$$, $$|vu|_a = 0$$ $$($$mod $$2)\}$$

where $$|vu|_a$$ is number of $$a$$ in $$vu$$.

I came up with these rules:

$$\sigma \rightarrow aa\sigma | ab\xi$$

$$\xi \rightarrow aa\xi | \epsilon$$

This generates words in form: $$(aa)^iab(aa)^j\epsilon$$, but not all words in $$L$$ look like this.

How do I make sure that $$b$$s can be put anywhere inbetween $$a$$s in $$v$$ and $$u$$.

Edit: I think I found a solution.

$$\sigma \rightarrow b\sigma$$ | $$a\alpha$$ | $$ab\gamma$$

$$\alpha \rightarrow b\alpha$$ | $$a\sigma$$ | $$ab\eta$$

$$\gamma \rightarrow b\gamma$$ | $$a\eta$$ | $$\epsilon$$

$$\eta \rightarrow b\eta$$ | $$a\gamma$$

"How do I make sure that $$b$$'s can be put anywhere in between $$a$$'s in v and u?" Hint, you may need another non-terminal.
Let us define two simpler languages, $$E=\{w | w\in \{a,b\}^*, |w|_a = 0(\text{mod } 2)\}$$ and $$O=\{w | w\in \{a,b\}^*, |w|_a = 1(\text{mod } 2)\}.$$
Can you write regular grammar for $$E$$ and $$O$$ ? Can you piece together some E's and O's to form $$L$$?