LR0, SLR1,LR1与LALR1分析的构造
前情提要
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A 代 表 非 终 结 符 A代表非终结符 A代表非终结符
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a 代 表 变 量 a代表变量 a代表变量
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项目集闭包
- 内核项: 初始项 S ′ → . S S' \rightarrow .S S′→.S以及
点不在最左端的所有项 - 非内核项: 除了初始项 S ′ → . S S' \rightarrow .S S′→.S以外
点在最左端的所有项
- 内核项: 初始项 S ′ → . S S' \rightarrow .S S′→.S以及
计算First集
- 算法开始 , 令 i = 1 i=1 i=1
- 判断X类型
- X是终结符, First(X)=X
- X是非终结符且 X → Y 1 Y 2 . . . Y n , n ≥ 1 X \rightarrow Y_1Y_2...Y_n, n \geq1 X→Y1Y2...Yn,n≥1
- if ε ∉ F i r s t ( Y i ) \varepsilon \notin First(Y_i) ε∈/First(Yi)
- F i r s t ( X ) = F i r s t ( X ) ∪ F i r s t ( Y i ) First(X) = First(X) \cup First(Y_i) First(X)=First(X)∪First(Yi)
- if ε ∈ F i r s t ( Y i ) \varepsilon \in First(Y_i) ε∈First(Yi)
- F i r s t ( X ) = F i r s t ( X ) ∪ { F i r s t ( Y i ) − { ε } } First(X) = First(X) \cup \lbrace First(Y_i)- \lbrace \varepsilon \rbrace \rbrace First(X)=First(X)∪{First(Yi)−{ε}}
- i = i + 1 i=i+1 i=i+1
- goto 算法开始
- if ε ∉ F i r s t ( Y i ) \varepsilon \notin First(Y_i) ε∈/First(Yi)
- X → ε X \rightarrow \varepsilon X→ε, 将 ε \varepsilon ε加入到First(X)中
- 算法结束
计算Follow集
- 将
#放入Follow(S)中, S为文法的开始符号,#为结束符 - 若存在 A → α B β A \rightarrow \alpha B \beta A→αBβ, 那么 F o l l o w ( B ) = F i r s t ( β ) − { ε } Follow(B) = First(\beta) - \lbrace \varepsilon \rbrace Follow(B)=First(β)−{ε}
- 若 A → α B A \rightarrow \alpha B A→αB或者 ( A → α B β A \rightarrow \alpha B \beta A→αBβ且 ε ∈ F i r s t ( β ) \varepsilon \in First(\beta) ε∈First(β)) , 那么 F o l l o w ( B ) = F o l l o w ( B ) ∪ F o l l o w ( A ) Follow(B) = Follow(B) \cup Follow(A) Follow(B)=Follow(B)∪Follow(A)
- 例子
- S → A B S \rightarrow AB S→AB
- A → ε ∣ a A \rightarrow \varepsilon|a A→ε∣a
- B → ε ∣ b B \rightarrow \varepsilon| b B→ε∣b
- First(S) = { a , b , ε } \lbrace a,b, \varepsilon \rbrace {a,b,ε}
LR(0)
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LR0_CLOSURE(I)
- LR0_CLOSURE(I)->Set_Items
- J = I;
- repeat
- for A → α . B β ∈ J A\rightarrow \alpha . B \beta \in J A→α.Bβ∈J
- for B → . γ B \rightarrow .\gamma B→.γ in G
- if B → . γ B \rightarrow .\gamma B→.γ not in J
- 将 B → . γ B \rightarrow .\gamma B→.γ加入J中
- if B → . γ B \rightarrow .\gamma B→.γ not in J
- for B → . γ B \rightarrow .\gamma B→.γ in G
- for A → α . B β ∈ J A\rightarrow \alpha . B \beta \in J A→α.Bβ∈J
- until 在某一轮中没有新的项加入到J中
- LR0_CLOSURE(I)->Set_Items
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LR0_Items(G’)
- LR0_Items(G’)
- C = {LR0_CLOSURE({[ S ′ → . S S' \rightarrow .S S′→.S]})}
- repeat
- for I in C
- for 每个文法符号X
- if G O T O ( I , X ) ≠ ∅ GOTO(I,X) \neq \emptyset GOTO(I,X)=∅ 且 不在C中
- GOTO(I,X)加入C中
- if G O T O ( I , X ) ≠ ∅ GOTO(I,X) \neq \emptyset GOTO(I,X)=∅ 且 不在C中
- for 每个文法符号X
- for I in C
- until 在某一轮中没有新的项加入到C中
- LR0_Items(G’)
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LR(0)分析表的构建
- 构造文法G’的项目集规范族 C = { I 0 , I 1 , I 2 . . . , I n } C = \lbrace I_0,I_1,I_2..., I_n\rbrace C={I0,I1,I2...,In}
- if [ A → α . a β ] ∈ I i [A\rightarrow \alpha .a \beta] \in I_i [A→α.aβ]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
sj - if [ S → S ′ . ] ∈ I i [S\rightarrow S'.] \in I_i [S→S′.]∈Ii, 则Action[i,
#]=Acc - if [ A → α . ] ∈ I i [A\rightarrow \alpha.] \in I_i [A→α.]∈Ii, 则for a ∈ T a \in T a∈T, 令Action[i,a]=r A → α . A\rightarrow \alpha. A→α.
- if G O T O ( I i , A ) = I j GOTO(I_i,A)=I_j GOTO(Ii,A)=Ij, 令GOTO[i,A]=j
- if [ A → α . a β ] ∈ I i [A\rightarrow \alpha .a \beta] \in I_i [A→α.aβ]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
- 构造文法G’的项目集规范族 C = { I 0 , I 1 , I 2 . . . , I n } C = \lbrace I_0,I_1,I_2..., I_n\rbrace C={I0,I1,I2...,In}
SLR(1)
- SLR(1)分析表的构建
- 构造文法G’的项目集规范族 C = { I 0 , I 1 , I 2 . . . , I n } C = \lbrace I_0,I_1,I_2..., I_n\rbrace C={I0,I1,I2...,In}
- if [ A → α . a β ] ∈ I i [A\rightarrow \alpha .a \beta] \in I_i [A→α.aβ]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
sj - if [ S → S ′ . ] ∈ I i [S\rightarrow S'.] \in I_i [S→S′.]∈Ii, 则Action[i,
#]=Acc - if [ A → α . ] ∈ I i [A\rightarrow \alpha.] \in I_i [A→α.]∈Ii, 则for a ∈ F o l l o w ( A ) a \in Follow(A) a∈Follow(A), 令Action[i,a]=r A → α . A\rightarrow \alpha. A→α.
- if G O T O ( I i , A ) = I j GOTO(I_i,A)=I_j GOTO(Ii,A)=Ij, 令GOTO[i,A]=j
其余与LR(0)相同
- if [ A → α . a β ] ∈ I i [A\rightarrow \alpha .a \beta] \in I_i [A→α.aβ]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
- 构造文法G’的项目集规范族 C = { I 0 , I 1 , I 2 . . . , I n } C = \lbrace I_0,I_1,I_2..., I_n\rbrace C={I0,I1,I2...,In}
LR(1)
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LR1_CLOSURE(I)
- repeat
- for ∀ [ A → α . B β , a ] ∈ I \forall[A \rightarrow \alpha.B \beta, a] \in I ∀[A→α.Bβ,a]∈I
- for ∀ [ B → γ ] ∈ G ′ \forall [B\rightarrow \gamma] \in G' ∀[B→γ]∈G′
- for ∀ b ∈ F i r s t ( β a ) \forall b \in First(\beta a) ∀b∈First(βa)
- 将 [ B → . γ , b ] [B \rightarrow .\gamma,b ] [B→.γ,b]加入集合I中
- for ∀ b ∈ F i r s t ( β a ) \forall b \in First(\beta a) ∀b∈First(βa)
- for ∀ [ B → γ ] ∈ G ′ \forall [B\rightarrow \gamma] \in G' ∀[B→γ]∈G′
- for ∀ [ A → α . B β , a ] ∈ I \forall[A \rightarrow \alpha.B \beta, a] \in I ∀[A→α.Bβ,a]∈I
- until
I中不能加入更多的项 - return I
- repeat
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LR1_GOTO(I,X)
- J = {}
- for ∀ [ A → α . X β , a ] ∈ J \forall [A \rightarrow \alpha.X \beta, a] \in J ∀[A→α.Xβ,a]∈J
- 将 [ A → α X . β , a ] [A \rightarrow \alpha X.\beta, a] [A→αX.β,a]加入集合J中
- return LR1_CLOSURE(J)
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items(G’)
- C = LR1_CLOSURE( { [ S ′ → . S , ♯ ] [S'\rightarrow .S,\sharp] [S′→.S,♯]})
- repeat
- for ∀ I ∈ C \forall I \in C ∀I∈C
- for 每个文法符号X
- if G O T O ( I , X ) ≠ ∅ GOTO(I,X) \neq \empty GOTO(I,X)=∅且不在C中
- GOTO(I,X)加入C中
- if G O T O ( I , X ) ≠ ∅ GOTO(I,X) \neq \empty GOTO(I,X)=∅且不在C中
- for 每个文法符号X
- for ∀ I ∈ C \forall I \in C ∀I∈C
- until 不再有新的项加入到
C中
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LR(1)分析表的构建
- 构造文法G’的LR(1)项目集规范族 C ′ = { I 0 , I 1 , I 2 . . . , I n } C' = \lbrace I_0,I_1,I_2..., I_n\rbrace C′={I0,I1,I2...,In}
- if [ A → α . a β , b ] ∈ I i [A\rightarrow \alpha .a \beta, b] \in I_i [A→α.aβ,b]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
sj - if [ S → S ′ . , ♯ ] ∈ I i [S\rightarrow S'., \sharp] \in I_i [S→S′.,♯]∈Ii, 则Action[i,
#]=Acc - if [ A → α . , a ] ∈ I i [A\rightarrow \alpha., a] \in I_i [A→α.,a]∈Ii, 则令Action[i,a]=r A → α . A\rightarrow \alpha. A→α.
- if G O T O ( I i , A ) = I j GOTO(I_i,A)=I_j GOTO(Ii,A)=Ij, 令GOTO[i,A]=j
- if [ A → α . a β , b ] ∈ I i [A\rightarrow \alpha .a \beta, b] \in I_i [A→α.aβ,b]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
- 构造文法G’的LR(1)项目集规范族 C ′ = { I 0 , I 1 , I 2 . . . , I n } C' = \lbrace I_0,I_1,I_2..., I_n\rbrace C′={I0,I1,I2...,In}
LALR(1)
传播法
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确定向前看符号:
look_ahead(K,X)- 输入: 项目集I的内核K, 文法符号X
- 算法描述
- for ∀ [ A → α . β ] ∈ K \forall [A \rightarrow \alpha.\beta] \in K ∀[A→α.β]∈K
- J = LR1_CLOSURE({[ A → α . β , ♯ A \rightarrow \alpha.\beta, \sharp A→α.β,♯]})
- if [ B → γ . X δ , a ] ∈ J [B \rightarrow \gamma.X \delta, a ] \in J [B→γ.Xδ,a]∈J
- 则 [ B → γ X . δ , a ] ∈ G O T O ( I , X ) [B \rightarrow \gamma X. \delta, a ] \in GOTO(I,X) [B→γX.δ,a]∈GOTO(I,X)中的
a是自发生成 - //build 自发生成表
- 则 [ B → γ X . δ , a ] ∈ G O T O ( I , X ) [B \rightarrow \gamma X. \delta, a ] \in GOTO(I,X) [B→γX.δ,a]∈GOTO(I,X)中的
- if [ B → γ . X δ , ♯ ] ∈ J [B \rightarrow \gamma.X \delta, \sharp ] \in J [B→γ.Xδ,♯]∈J
- 向前看符号从I中 A → α . β A \rightarrow \alpha.\beta A→α.β传播到GOTO(I,X)中的项 B → γ X . δ B \rightarrow \gamma X. \delta B→γX.δ之上
- //build 传播表
- for ∀ [ A → α . β ] ∈ K \forall [A \rightarrow \alpha.\beta] \in K ∀[A→α.β]∈K
- 输出: 自发生成表, 传播表
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计算LALR(1)项目集族内核
- let C = G的LR(0)项目集族 , 并删除其中的非内核项
- let table = {}; let broadcast = {}
- for ∀ X ∈ V N ∪ V T \forall X \in V_N \cup V_T ∀X∈VN∪VT
- for ∀ K ∈ C \forall K \in C ∀K∈C
- let tmp_table, tmp_broadcast = look_ahead(K,X)
- table.extend(tmp_table);
- broadcast.extend(tmp_broadcast);
- for ∀ K ∈ C \forall K \in C ∀K∈C
- 传播
- for ∀ ( i t e m i , i t e m j s ) ∈ b r o a d c a s t \forall (item_i, item_js ) \in broadcast ∀(itemi,itemjs)∈broadcast
- for ∀ i t e m j ∈ i t e m j s \forall item_j \in item_js ∀itemj∈itemjs
- 把 i t e m i item_i itemi的展望符传播到 i t e m j item_j itemj中去
- for ∀ i t e m j ∈ i t e m j s \forall item_j \in item_js ∀itemj∈itemjs
- for ∀ ( i t e m i , i t e m j s ) ∈ b r o a d c a s t \forall (item_i, item_js ) \in broadcast ∀(itemi,itemjs)∈broadcast
-
LALR(1)分析表的构建
- 构造文法G’的LALR(1)项目集规范族 C ′ = { I 0 , I 1 , I 2 . . . , I n } C' = \lbrace I_0,I_1,I_2..., I_n\rbrace C′={I0,I1,I2...,In}
- if [ A → α . a β , b ] ∈ I i [A\rightarrow \alpha .a \beta, b] \in I_i [A→α.aβ,b]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
sj - if [ S → S ′ . , ♯ ] ∈ I i [S\rightarrow S'., \sharp] \in I_i [S→S′.,♯]∈Ii, 则Action[i,
#]=Acc - if [ A → α . , a s ] ∈ I i [A\rightarrow \alpha., as] \in I_i [A→α.,as]∈Ii, 则for ∀ a ∈ a s \forall a \in as ∀a∈as: then Action[i,a]=r A → α . A\rightarrow \alpha. A→α.
- //
as是展望符的集合
- //
- if G O T O ( I i , A ) = I j GOTO(I_i,A)=I_j GOTO(Ii,A)=Ij, 令GOTO[i,A]=j
- if [ A → α . a β , b ] ∈ I i [A\rightarrow \alpha .a \beta, b] \in I_i [A→α.aβ,b]∈Ii并且 G O T O ( i , a ) = j GOTO(i,a)=j GOTO(i,a)=j, 则Action[i,a]=
- 构造文法G’的LALR(1)项目集规范族 C ′ = { I 0 , I 1 , I 2 . . . , I n } C' = \lbrace I_0,I_1,I_2..., I_n\rbrace C′={I0,I1,I2...,In}
参考
龙书