Formal Languages and Compilers-LL(1),FIRST and FOLLOW

※递归下降法：简而言之，就是比如文法是S→T，T→FM, F→+M，M→a用伪代码表示这个文法则首先要写 procedure S BEGINT;END然后写 procedure TBEGINF;M;ENDProcedure FBEGIN+M;ENDProcedure MBEGIN匹配aEND这样的过程是递归下降的过程。递归下降方法仅表示一种思路，没有特别的

※LL(1)预测分析法：第一个L表示从左到右扫描输入串第二个L表示最左推导1表示分析时每步只需向前查看一个符号

How to identify whether a grammar is LL(1), LR(0) or SLR(1)?

If you have no FIRST/FIRST conflicts and no FIRST/FOLLOW conflicts, your grammar is LL(1).An example of a FIRST/FIRST conflict:

S -> Xb | Yc
X -> a
Y -> a

By seeing only the first input symbol a, you cannot know whether to apply the production S -> Xb or S -> Yc, because a is in the FIRST set of both X and Y.An example of a FIRST/FOLLOW conflict:

S -> AB
A -> fe | epsilon
B -> fg

By seeing only the first input symbol f, you cannot decide whether to apply the production A -> fe or A -> epsilon, because f is in both the FIRST set of A and the FOLLOW set of A (A can be parsed as epsilon and B as f).Notice that if you have no epsilon-productions you cannot have a FIRST/FOLLOW conflict.
2.X → Yz | aY → bZ | εZ → εTo check if a grammar is LL(1), one option is to construct the LL(1) parsing table and check for any conflicts. These conflicts can beFIRST/FIRST conflicts, where two different productions would have to be predicted for a nonterminal/terminal pair.

FIRST/FOLLOW conflicts, where two different productions are predicted, one representing that some production should be taken and expands out to a nonzero number of symbols, and one representing that a production should be used indicating that some nonterminal should be ultimately expanded out to the empty string.

FOLLOW/FOLLOW conflicts, where two productions indicating that a nonterminal should ultimately be expanded to the empty string conflict with one another.

Let's try this on your grammar by building the FIRST and FOLLOW sets for each of the nonterminals. Here, we get that

FIRST(X) = {a, b, z}
FIRST(Y) = {b, epsilon}
FIRST(Z) = {epsilon}

We also have that the FOLLOW sets are

FOLLOW(X) = {$}
FOLLOW(Y) = {z}
FOLLOW(Z) = {z}

From this, we can build the following LL(1) parsing table:

a    b    z   $
X   a    Yz   Yz
Y        bZ   eps
Z             eps

Since we can build this parsing table with no conflicts, the grammar is LL(1).To check if a grammar is LR(0) or SLR(1), we begin by building up all of the LR(0) configurating sets for the grammar. In this case, assuming that X is your start symbol, we get the following:

(1)
X' -> .X
X -> .Yz
X -> .a
Y -> .
Y -> .bZ

(2)
X' -> X.

(3)
X -> Y.z

(4)
X -> Yz.

(5)
X -> a.

(6)
Y -> b.Z
Z -> .

(7)
Y -> bZ.

From this, we can see that the grammar is not LR(0) because there are shift/reduce conflicts in states (1) and (6). Specifically, because we have the reduce items Z → . and Y → ., we can't tell whether to reduce the empty string to these symbols or to shift some other symbol. More generally, no grammar with ε-productions is LR(0).However, this grammar might be SLR(1). To see this, we augment each reduction with the lookahead set for the particular nonterminals. This gives back this set of SLR(1) configurating sets:

(1)
X' -> .X
X -> .Yz [$]
X -> .a  [$]
Y -> .   [z]
Y -> .bZ [z]

(2)
X' -> X.

(3)
X -> Y.z [$]

(4)
X -> Yz. [$]

(5)
X -> a.  [$]

(6)
Y -> b.Z [z]
Z -> .   [z]

(7)
Y -> bZ. [z]

Now, we don't have any more shift-reduce conflicts. The conflict in state (1) has been eliminated because we only reduce when the lookahead is z, which doesn't conflict with any of the other items. Similarly, the conflict in (6) is gone for the same reason.