Known and described subclasses of Context-Free Grammars class

I'm looking for various researches which consider specific subclasses of Context-Free Grammar class, i.e. some specific described cases, which differ from well-known:

• deterministic/non-deterministic
• ambiguous/unambiguous
• regular/non-regular

As an example of such "non-standard" subclasses are Visibly pushdown grammar described here. Any additional examples will be much appreciated.

More specifically, I'm wondering, are any described subclasses which can help distinguish this two very similar cases?

$G_1$:

$S\rightarrow aD$, $S\rightarrow cD$

$D\rightarrow abc$, $D\rightarrow abDc$

$G_2$:

$S\rightarrow aD$, $S\rightarrow cD$

$D\rightarrow abc$, $D\rightarrow abDc$, $D\rightarrow aDb$, $D\rightarrow aDc$

It is clear that both of them are unambiguous non-determ. CFG, and $\mathcal{L}(G_1)\subset\mathcal{L}(G_2)$. But these are common properties, I'm looking for specific differences.

• Aside: are you sure that $G_1$ is non-deterministic? Why can't a $LR(0)$ parser reduce $abc$ or $abDc$ to $D$ every time it shifts a $c$? – gdmclellan Jun 26 '16 at 21:55
• Sorry, why it should (or can) shift c? $abDc$ makes this grammar non-regular (produces $(ab)^k abc (c)^k$). – Andrey Lebedev Jun 26 '16 at 23:08
• I agree that the language of $D$ isn't regular. However, deterministic languages are a much broader class, encompassing all the $LR$ languages. – gdmclellan Jun 26 '16 at 23:43

Density might be interesting concept for you. The density function is defined as $$\delta_L(n) := |L\cap \Sigma^n|,$$
where $\Sigma^n$ denotes the set of all strings of length $n$ over $\Sigma$.
It might be worth looking at your example in terms of the Chomsky-Schützenberger theorem. The theorem's statement for the two grammars in question is pretty trivial, but you'll probably find that the brace-pair alphabet for the smallest Dyck language of which $L(G_1)$ is a homomorphic image is smaller than the that for $L(G_2)$. (At a glance, I'm counting 3 and 5 respectively.)