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：

E -> TE’

E’ -> +TE’ | ε 

T -> FT’

T’ -> *FT’ | ε 

F -> (E) | i

 

分析符号串 i*i+i：

FIRST集：		FOLLOW集：
FIRST(E) = { ( , i } FIRST(E’) = {+ , ε } FIRST(T) = { ( , i } FIRST(T’) = { * , ε } FIRST(F) = { ( , i }	FIRST(TE') = { (, i } FIRST(+TE') = {+} FIRST(ε) = {ε} FIRST(FT') = { (, i } FIRST(*FT') = {*} FIRST((E)) = { ( } FIRST(i) = {i}	FOLLOW(E) = { ) , # } FOLLOW(E’) = { ) , # } FOLLOW(T) = { + , ) ,# } FOLLOW(T’) = { + , ) ,# } FOLLOW(F) = { * , + , ) ,# }
SELECT集：
SELECT(E -> TE’) = FIRST(E) = { ( , 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

 

 P101练习7（2）

 

A -> aABe | a

 

B -> Bb | d

 

文法改写：

 

A -> aA’

 

A’ -> ABe | ε

 

B -> dB’

 

B’ -> bB’ | ε

 

FIRST集：		FOLLOW集：
FIRST(A) = { a } FIRST(A’) = { a , ε } FIRST(B) = { d } FIRST(B’) = { b , ε }	FIRST(aA')={a} FIRST(ABe)={a} FIRST(ε)={ε} FIRST(dB')={d} FIRST(bB')={b}	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) = FIRST(ε) - {ε} U FOLLOW(A’) = FOLLOW(A’) = { d , # } SELECT(B -> dB’) = FIRST(dB’) = { d } SELECT(B’ -> bB’) = FIRST(bB’) = {b } SELECT(B’ -> ε) = FIRST(ε) - {ε} U FOLLOW(B’) = FOLLOW(B’) = { e }

 

 

 P101练习7（3）

 

S -> SBa | b

 

B -> ab

 

文法改写：

 

S -> bS’

 

S’-> BaS’| ε

 

B -> ab

 

FIRST集：		FOLLOW集：
FIRST(S) = { b } FIRST(S’) = { a , ε } FIRST(B) = { a }	FIRST(bS')={b} FIRST(BaS')={a} FIRST(ε)={ε} FIRST(ab)={a}	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
A->cA

A->aA

 

 

 

FIRST集：		FOLLOW集：
FIRST(S) = { a,c,p } FIRST(A) = { a,c,ε }	FIRST(Ap)={a,c,p} FIRST(a)={a} FIRST(ε)={ε} FIRST(cA)={c} FIRST(aA)={a}	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->cA
B->b
B->dB

 

 

FIRST集：		FOLLOW集：
FIRST(S1) = { a , c } FIRST(S2) = { b , d } FIRST(A) = { a , c } FIRST(B) = { b ,d }	FIRST(Ap)={a,c} FIRST(Bq)={b,d} FIRST(a)={a} FIRST(cA)={c} FIRST(b)={b} FIRST(dB)={d}	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 }