编译原理：消除左递归

1.将以下文法消除左递归，分析符号串 i*i+i 。

   并分别求FIRST集、FOLLOW集，和SELECT集

     E -> E+T | T

　　E -> TE'  

　　E' -> +TE'|ε

 

     T -> T*F | F

　　T -> FT'

　　T' -> *F|ε

 

     F -> (E) | i

 解析：

1.First集：

First(TE') = {T}

First(+TE') = {+}

First(ε) = {ε}

First(FT') = {F}

First(*F) = {*}

First((E)) = {(}

First(i) = {i}

2.Follow集：

Follow(E) = {)}

Follow(E') = {ε}

Follow(T) = {E'}

Follow(T') ={ε}

Follow(F) = {ε}

3.Select集：

Select(E -> TE') = First(TE') = {T}

Select(E' -> +TE') = First(+TE') = {+}

Select(E' -> ε) = (First(ε)-{ε})∪Follow(E') = {)}

Select(T -> FT') = First(FT') = {F}

Select(T' -> *F) = First(*F) = {*}

Select(T' -> ε) = (First(ε)-{ε})∪Follow(T') = {ε}

Select(F -> (E)) = First((E)) = {(}

Select(F -> i ) = First(i) = {i}

2.P101练习7（2）（3）文法改写，并分别求FIRST集、FOLLOW集，和SELECT集

(2)

　　A -> aABe|a

　　B -> Bb|d

解析：

提取左公因子：

　　A→aA'

　　A'→ABe|ε

消除公因式：

　　B→dB'

　　B'→bB'|ε

 

1.First集：

First(aA') = {a}

First(ABe) = {a}

First(ε) = {ε}

First(dB') = {d}

First(bB') = {b}

2.Follow集：

Follow(A) = {Be}

Follow(A') = {ε}

Follow(B) = {e}

Follow(B') = {ε}

3.Select集：

Select(A -> aA') = First(aA') = {a}

Select(A' -> ABe) = First(ABe) = {a}

Select(A' -> ε) =  (First(ε)-{ε})∪Follow(A')) = {ε}

Select(B -> dB'）= First(dB') = {d}

Select(B' -> bB') = First(bB') = {b}

Select(B' -> ε) = (First(ε)-{ε})∪Follow(B')) = {ε}

 (3)

　　S -> Aa|b

　　A ->SB

　　B ->ab

②代入①：

　　S -> SBa|b

消除左递归：

　　S - > bS'

　　S' -> BaS'|ε

　　B -> ab

 

1.First集：

First(SBa) = {a}

First(b) = {b}

First(bS') = {b}

First(BaS') = {a}

First(ε) = {ε}

First(ab) = {a}

2.Follow集：

Follow(S) = {B}

Follow(S') = {ε}

Follow(B) = {S'}

3.Select集：

Select(S -> SBa) = First(SBa) = {a}

Select(S -> b) = First(b) = {b}

Select(S -> bS') = First(bS') = {b}

Select(S' -> BaS') = First(BaS') = {a}

Select(S' -> ε) =  (First(ε)-{ε})∪Follow(S')) = {ε}

Select(B -> ab) = First(ab) = {a}

课堂练习：求以下文法的FIRST集、FOLLOW集和SELECT集。

（1）

　　S->Ap
　　A->a |ε
　　A->cA

　　A->aA

 解析：

1.First集：

First(S) = {a,c,p}

First(a) = {a}

First(ε) = {ε}

First(cA) = {c}

FIrst(aA) = {a}

2.Follow集：

Follow(S) = {ε}

Follow(A) ={p}

3.Select集：

Select(S->Ap) = First(Ap) ={p}

Select(A -> a) = FIrst(a) = {a}

Select(A -> ε) = First(ε) = {ε}

Select(A -> cA) = First(cA) ={c}

Select(A -> aA) =First(aA) = {a}

（2）

　　S->Ap
　　S->Bq
　　A->a
　　A->cA
　　B->b
　　B->dB

解析：

1.First集：

First(Ap) = {a,c}

First(Bq) = {b,d}

First(a) = {a}

First(cA) = {c}

First(b) = {b}

First(dB) = {d}

2.Follow集：

Follow(S) = {ε}

Follow(A) = {p}

Follow(B) = {q}

3.Select集：

Select(S -> Ap) = First(Ap) = {a,c}

Select(S - >Bq) = First(Bq) = {q}

Select(A - > a) = First(a) = {a}

Select(A -> cA) = First(cA) = {c}

Select(B -> b）= FIrst(b) = {b}

Select(B ->dB) = FIrst(dB) = {d}