编译原理10 消除左递归
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
T -> T*F | F
F -> (E) | i
消除左递归得：
E->TE’
E’->+TE’|ε
T->FT’
T’->*FT’|ε
F->(E)|i
分析字符串如下：
FIRST集：
FIRST(TE’)={ (, i }
FIRST(+TE')={+，ε}
FIRST(FT’)={(,i}
FIRST(*FT')={ *,ε}
FIRST((E))={ (}
FIRST(i)={ i }
FOLLOW集：
FOLLOW(E)={ ),# }
FOLLOW(E')={ ),# }
FOLLOW(T)={+,),#}
FOLLOW(T')={+,),#}
FOLLOW(F)={*,+,),#}
SELECT集：
SELECT (E -> TE’) = FIRST(TE’) = { ( , i }
SELECT(E’ -> +TE’) = FIRST(+TE’) = { + }
SELECT(E’ -> ε) = FIRST(ε) - {ε} U FOLLOW(E’) = FOLLOW(E’) = { ) , # }
SELECT(T -> FT’) = FIRST(FT’) = { ( , i }
SELECT(T’ -> *FT’) = FIRST(*FT’) = { * }
SELECT(T’ -> ε) = FIRST(ε) - {ε} U FOLLOW(T’) = FOLLOW(T’) = { + , ) ,# }
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
原题：
(1)A->aABe|a
B->Bb|d
消除左递归得：
A -> aA’
A’ -> ABe | ε
B -> dB’
B’ -> bB’ | ε
FIRST集：
FIRST(aA')={a}
FIRST(ABe)= FIRST(aA’Be)={a}
FIRST(ε)={ε}
FIRST(dB')={d}
FIRST(bB')={b}
FOLLOW集：
FOLLOW(A)={d,#}
FOLLOW(A')={d,#}
FOLLOW(B)={e,#}
FOLLOW(B')={e,#}
SELECT集：
SELECT(A->aA')=FIRST(aA')={a}
SELECT(A'->ABe)=FIRST(ABe)={a}
SELECT(A'->ε) =FOLLOW(A')={d,#}
SELECT(B->dB')=FIRST(dB')={d}
SELECT(B'->bB')=FIRST(bB')={b}
SELECT(B'->ε) =FOLLOW(B')={e}
(3)S->Aa|b
A->SB
B->ab
S -> SBa | b
B -> ab
文法改写：
S -> bS’
S’ -> BaS’ | ε
B -> ab
FIRST集：
FIRST(bS’) = { b }
FIRST(BaS’) = { a , ε }
FIRST(ab) = { a }
FOLLOW集：
FOLLOW(S) = {#}
FOLLOW(S’) = {#}
FOLLOW(B) = {a}
SELECT集：
SELECT (S -> bS’) = FIRST(bS’) = { b }
SELECT(S’ -> BaS’) = FIRST(BaS’) = { a }
SELECT(S’ -> ε) = FIRST(ε) - {ε} U FOLLOW(S’) = FOLLOW(S’) = { # }
SELECT(B -> ab) = FIRST(ab) = { a }
课堂练习：
求以下文法的FIRST集、FOLLOW集和SELECT集。
S->Ap
A->a |ε # 有ε空 可跳P
A->cA
A->aA
FIRST集：
FIRST(AP) = { a , c , p }
FIRST(A) = { a , c , ε }
FOLLOW集：
FOLLOW(S) = { # }
FOLLOW(A) = { p }
SELECT集：
SELECT(S -> Ap) = FIRST(Ap) = { a , c , p }
SELECT(A -> a ) = FIRST(a) = { a }
SELECT(A -> ɛ) = FIRST(ε) - {ε} U FOLLOW(A) = FOLLOW(A) = { p }
SELECT(A -> cA) = FIRST(cA) = { c }
SELECT(A -> aA) = FIRST(aA) = { a }
S->Ap
S->Bq
A->a #A无 ε不可跳P
A->cA
B->b
B->dB
FIRST集：
FIRST(S1) = FIRST(Ap) = { a , c }
FIRST(S2) = FIRST(Bq) = { b , d }
FIRST(A) = { a , c}
FIRST(B) = { b ,d }
FOLLOW集：
FOLLOW(S) = { # }
FOLLOW(A) = { p }
FOLLOW(B) = { q }
SELECT集：
SELECT(S -> Ap) = FIRST(Ap) = { a , c }
SELECT(S -> Bq ) = FIRST(Bq) = { b , d }
SELECT(A -> a) = FIRST(a) = { a }
SELECT(A -> cA) = FIRST(cA) = { c }
SELECT(B -> b) =FIRST(b) = { b }
SELECT(B -> dB) = FIRST(dB) = { d }
