[Compilation Principle] Formal Language Classification

to sum up

This is based on "Compilation Principles Tutorial (Fourth Edition)" Hu Yuanyi
(I have read other teachers, and they are a bit different . I will explain the different points in the points of attention)

The difference between the 4 types of grammar

Type 0

The only restriction : there must be a non-terminal symbol in the left part of the production

Type 1

The only difference with Type 0 : the right part of the production cannot be empty (in order to ensure that the length of the right part is not shorter than the length of the left)

Type 2

The only restriction : the left part of the production has exactly one nonterminal

Type 3

The only difference with Type 2 : There is at most one non-terminal symbol on the right of the production

important point

About Type 2

$$\alpha A\delta \to \alpha \beta \delta$$
$A$ must be in$\alpha$、$In$ the context of$\delta$ can it beβ \betareplaced by$\beta$ , which is certain

Then the book says that there must be at least one non-terminal symbol in the right part of the production, that is, the type 1 grammar cannot appear $S\to$A production in the form of$\epsilon$ .

There is another saying:
(1) If there is $S\to \epsilon$ , then$S$ must not appear in the right part of the production
(2) generally not allowed to be replaced with$\epsilon$ unless the start symbol produces$\epsilon$
Then this statement, the right part of the production allows no non-terminal symbols
[the two are essentially the same:$|\alpha |$ $\le$ $|\beta |$】

If you follow this statement, then the relationship between grammars is:

otherwise, as the summary table says, type 2 grammars may not necessarily belong to type 1 grammars , because type 2 grammars may have $S\to e$

What kind of statement is based on what the teacher teaches.