自底向上语法分析LR(1)

自底向上分析方法，也称移进－归约分析法，粗略地说它的实现思想是对输入符号串自左向右进行扫描，并将输入符逐个移入一个后进先出栈中，边移入边分析，一旦栈顶符号串形成某个句型的句柄时，就用该产生式的左部非终结符代替相应右部的文法符号串，这称为归约。重复这一过程直到归约到栈顶中只剩文法的开始符号时则为分析成功，也就确认输入串是文法的句子。

概述

设文法G[S]为：
（1）S→aAcBe
（2）A→b
（3）A→Ab
（4）B→d

对输入串abbcde#进行分析，检查该符号串是否是G[S]的句子
由于自底向上分析的移进－归约过程是自顶向下最右推导的逆过程，而最右推导为规范推导，自左向右的归约过程也称为规范归约。

步骤	符号栈	输入符串	动作
1	#	abbcde#	移进
2	#a	bbcde#	移进
3	#ab	bcde#	归约（A->b）
4	#aAb	cde#	归约（A->Ab）
5	#aAc	de#	移进
6	#aAcd	e#	归约（B->d）
7	#aAcBe	#	归约（S→aAcBe）
9	#S#		接受

在上述移进－归约或自底向上构造语法树的过程中，考虑几个问题：

何时移进？
何时归约？
将哪个字符串归约？

当一个文法无二义性时，那么它对一个句子的规范推导是唯一的，规范规约也必然是唯一的。因而每次归约时要找当前句型的句柄，也就是说，任何时候栈中的符号串和剩余的输入串组成一个句型，当句柄出现在栈顶符号串中时，则可用句柄归约，这样一直归约到输入串只剩结束符，文法符号栈中只剩开始符号。

由此可见，自底向上分析的关键问题是在分析过程中如何确定句柄，即如何知道何时在栈顶符号串中已形成某句型的句柄。然而自底向上的分析算法很多，目前常用的算符优先分析和LR类分析法。

直观算符优先分析法

给定文法G[E]：
E → E + E
E -> E - E
E -> E * E
E -> E / E
E -> E ↑ E 表示次方2↑3↑2 = 2↑9 = 512
E -> ( E )
E -> i

1.规定算法的优先关系如下：

1）↑ 优先级最高，具有右结合性（ ↑ < ↑ ）【注：表示次方2↑3↑2 = 2↑9 = 512】
2）*和/优先级低于↑，具有左结合性
3）+和-优先级最低，具有左结合性
4）(和)，括号的优先级小于括号内运算符，大于括号外运算符优先级；内括号的优先级大于外括号的优先级
5）任何于#相邻的符号的优先级都大于#

2.算符优先关系表如下

	+	-	*	/	↑	(	)	i	#
+	>	>	<	<	<	<	>	<	>
-	>	>	<	<	<	<	>	<	>
*	>	>	>	>	<	<	>	<	>
/	>	>	>	>	<	<	>	<	>
↑	>	>	>	>	<	<	>	<	>
(	<	<	<	<	<	<	=	<
)	>	>	>	>	>		>		>
i	>	>	>	>	>		>		>
#	<	<	<	<	<	<		<	=

3.输入字符串i + i * i

步骤	符号栈	当前输入符	输入符串	动作
1	#	i	+i*i#	移进(#<i)
2	#i	+	i*i#	归约(i>+)
3	#E	+	i*i#	移进(#<+)
4	#E+	i	*i#	移进(+<i)
5	#E+i	*	i#	归约(i>*)
6	#E+E	*	i#	移进(+>*)
7	#E+E*	i	#	移进（*<i）
8	#E+E*i	#		归约(i>#)
9	#E+E*E	#		归约(*>#)
10	#E+E	#		归约(+>#)
11	#E	#		acc

算符优先分析法

算符优先分析法(Operator Precedence
Parse)是仿效四则运算的计算过程而构造的一种语法分析方法。算符优先分析法的关键是比较两个相继出现的终结符的优先级而决定应采取的动作。

给文法一点的限制，把符号优先线顺序反应在文法中，再用方法求出其算符优先关系表。详情（略)

LR(0)分析

给定文法G[E]：
(1) E -> aA|bB
(2) A -> cA|d
(3) B->cB|d
G[E]的拓广文法，得到G [S’]：
(0) S’ -> E
(1) E -> aA|bB
(2)A -> cA|d
(3) B->cB|d

构造G[S’] 的 LR(0) FSM

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I0:
S'->.E
E->.aA
E->.bB

I1:
S'->E.

I2:
E->a.A
A->.cA
E->.d

I3:
E->b.B
B->.cB
B->.d

I6:
E->aA.

I4:
A->c.A
A->.cA
A->.d

I10:
E->d.

I7
E->bB.

I5
B->c.B
B->.cB
B->.d

I11
B->d.

I8:
A->cA.

I9:
B->cB.

LR(0)分析表

移进项目：如A->α•aβ,圆点后面为终结符的项目为移进项目

待约项目：如A->α•Bβ,圆点后面为非终结符的项目为待约项目

归约项目：如A->α•,圆点在最右端的项目为归约项目

接受项目：如S’->α•,S’->α为拓广文法，S’为左部的产生式只有一个，因此为特殊的归约项目，为接受项目

ACTION 表项和 GOTO表项可按如下方法构造(lk表示FSM图中的状态，中可以0,1,2…)：
若项目A ->α • aβ属于 Ik 且 GO (Ik, a)= Ij, 期望字符a 为终结符，则置ACTION[k, a] =sj(j表示新状态Ij);
若项目A ->α • Aβ属于Ik,且GO (Ik, A)= Ij,期望字符 A为非终结符，则置GOTO(k, A)=j (j表示文法中第j个产生式);
若项目A ->α •属于Ik, 那么对任何终结符a, 置ACTION[k, a]=rj；其中，假定A->α为文法G 的第j个产生式；
若项目S’ ->S • 属于Ik, 则置ACTION[k, #]为“acc”;
分析表中凡不能用上述规则填入信息的空白格均置上“出错标志”

状态	ACTION					GOTO
状态	a	b	c	d	#	E	A	B
0	S2	S3				1
1					acc
2			S4	S10			6
3			S5	S11				7
4			S4	S10			8
5			S5	S11				9
6	r1	r1	r1	r1	r1
7	r2	r2	r2	r2	r2
8	r3	r3	r3	r3	r3
9	r5	r5	r5	r5	r5
10	r4	r4	r4	r4	r4
11	r6	r6	r6	r6	r6

文法 G 是 LR(0) 文法，当且仅当它的LR(0)FSM中的每个状态都满足：
①不同时含有移进项目和归约项目，即不存在移进-归约冲突。
②不含有两个以上归约项目，即不存在归约-归约冲突。

对输入串bccd#的分析过程

步骤	状态栈	符号栈	输入符串	ACTION	GOTO
1	0	#	bccd#	S3
2	03	#b	ccd#	S5
3	035	#bc	cd#	S5
4	0355	#bcc	d#	S11
5	0355(11)	#bccd	#	r6	9
6	03559	#bccB	#	r5	9
7	0359	#bcB	#	r5	7
8	037	#bB	#	r2	1
9	01	#E	#	acc

碰到r的时候，状态栈出栈【r式子右部长度】然后根据符号栈入栈【r式子左边字符】

SLR(1)分析

LR(0)文法不能有移进-归约冲突及归约-归约冲突的限制。大多程序设计的时候不能满足。为了解决冲突，我们当冲突发生的时候向前查看一个符号，以确定哪个动作，这种分析方法为简单的LR(1)，称为SLR(1).

详情(略)…

LR(1)分析

SLR(1)分析法只是简单地考察下一个输入符号a是否属于归约项目A→α相关联的FOLLOW(A)，就采用产生式A->α归约，但 a∈FOLLOW(A) 只是归约的一个必要条件，而非充分条件。如果栈里的符号串是βα，归约成βA,再移当前符a,栈里变为βAa,而实际上βAa不一定是文件的活前缀。

A→α规约栈顶的符号α时，不仅应该查看栈中符号串δα，还应向前扫视一个输入符号a，只有当δAa确实构成文法某一规范句型的前缀时，才能用此规则进行规约。

给定文法G[S’]：
S’->S
S->aAd
S->bAc
S->aec
S->bed
A->e

LR(1)项目集族的构造

以S′→·S，#属于初始项目集中，把’#‘号作为向前搜索符，表示活前缀为γ(若γ是有关S产生式的某一右部)要归约成S时，必须面临输入符为’#'号才行。我们对初始项目S′→·S，# 求闭包后再用转换函数逐步求出整个文法的LR(1)项目集族。具体构造步骤如下：
a)假定I是一个项目集， I 的任何项目都属于CLOSURE(I)。
b) 若有项目 A→α·Bβ,a属于CLOSURE(I)，B→γ是文法中的产生式，β∈V*，b∈FIRST(βa)，则 B→·γ,b 也属于CLOSURE(I)中。
c) 重复b)直到CLOSURE(I)不再增大为止。

1.我们对初始项目S′→·S，#

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I0:
S'->S,#

2.活前缀为γ(若γ是有关S产生式的某一右部)要归约成S时，必须面临输入符为’#'号才行

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.box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-JDlSvaeomBcxiSFm .edgeLabel text{fill:#333}#mermaid-svg-JDlSvaeomBcxiSFm .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-JDlSvaeomBcxiSFm .node circle.state-start{fill:black;stroke:black}#mermaid-svg-JDlSvaeomBcxiSFm .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-JDlSvaeomBcxiSFm #statediagram-barbEnd{fill:#9370db}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-state .divider{stroke:#9370db}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-JDlSvaeomBcxiSFm .note-edge{stroke-dasharray:5}#mermaid-svg-JDlSvaeomBcxiSFm .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-JDlSvaeomBcxiSFm .error-icon{fill:#522}#mermaid-svg-JDlSvaeomBcxiSFm .error-text{fill:#522;stroke:#522}#mermaid-svg-JDlSvaeomBcxiSFm .edge-thickness-normal{stroke-width:2px}#mermaid-svg-JDlSvaeomBcxiSFm .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-JDlSvaeomBcxiSFm .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-JDlSvaeomBcxiSFm .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-JDlSvaeomBcxiSFm .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-JDlSvaeomBcxiSFm .marker{fill:#333}#mermaid-svg-JDlSvaeomBcxiSFm .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-JDlSvaeomBcxiSFm {color: rgba(0, 0, 0, 0.75);font: ;}

I0:
S'->S,#
S->.aAd,#
S->.bAc,#
S->.aec,#
S->.bed,#

3.go(I0,a)
若有项目 S->a.Ad,#属于I2，A→e是文法中的产生式，{d}∈V*，d∈FIRST({d#})，则 A→·e,d 也属于I2中

#mermaid-svg-vhE0KWMCE0KhqTAy .label{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);fill:#333;color:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .label text{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .node rect,#mermaid-svg-vhE0KWMCE0KhqTAy .node circle,#mermaid-svg-vhE0KWMCE0KhqTAy .node ellipse,#mermaid-svg-vhE0KWMCE0KhqTAy .node polygon,#mermaid-svg-vhE0KWMCE0KhqTAy .node path{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-vhE0KWMCE0KhqTAy .node .label{text-align:center;fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .node.clickable{cursor:pointer}#mermaid-svg-vhE0KWMCE0KhqTAy .arrowheadPath{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .edgePath .path{stroke:#333;stroke-width:1.5px}#mermaid-svg-vhE0KWMCE0KhqTAy .flowchart-link{stroke:#333;fill:none}#mermaid-svg-vhE0KWMCE0KhqTAy .edgeLabel{background-color:#e8e8e8;text-align:center}#mermaid-svg-vhE0KWMCE0KhqTAy .edgeLabel rect{opacity:0.9}#mermaid-svg-vhE0KWMCE0KhqTAy .edgeLabel span{color:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .cluster rect{fill:#ffffde;stroke:#aa3;stroke-width:1px}#mermaid-svg-vhE0KWMCE0KhqTAy .cluster text{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:12px;background:#ffffde;border:1px solid #aa3;border-radius:2px;pointer-events:none;z-index:100}#mermaid-svg-vhE0KWMCE0KhqTAy .actor{stroke:#ccf;fill:#ECECFF}#mermaid-svg-vhE0KWMCE0KhqTAy text.actor>tspan{fill:#000;stroke:none}#mermaid-svg-vhE0KWMCE0KhqTAy .actor-line{stroke:grey}#mermaid-svg-vhE0KWMCE0KhqTAy .messageLine0{stroke-width:1.5;stroke-dasharray:none;stroke:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .messageLine1{stroke-width:1.5;stroke-dasharray:2, 2;stroke:#333}#mermaid-svg-vhE0KWMCE0KhqTAy #arrowhead path{fill:#333;stroke:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .sequenceNumber{fill:#fff}#mermaid-svg-vhE0KWMCE0KhqTAy #sequencenumber{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy #crosshead path{fill:#333;stroke:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .messageText{fill:#333;stroke:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .labelBox{stroke:#ccf;fill:#ECECFF}#mermaid-svg-vhE0KWMCE0KhqTAy .labelText,#mermaid-svg-vhE0KWMCE0KhqTAy .labelText>tspan{fill:#000;stroke:none}#mermaid-svg-vhE0KWMCE0KhqTAy .loopText,#mermaid-svg-vhE0KWMCE0KhqTAy .loopText>tspan{fill:#000;stroke:none}#mermaid-svg-vhE0KWMCE0KhqTAy .loopLine{stroke-width:2px;stroke-dasharray:2, 2;stroke:#ccf;fill:#ccf}#mermaid-svg-vhE0KWMCE0KhqTAy .note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-vhE0KWMCE0KhqTAy .noteText,#mermaid-svg-vhE0KWMCE0KhqTAy .noteText>tspan{fill:#000;stroke:none}#mermaid-svg-vhE0KWMCE0KhqTAy .activation0{fill:#f4f4f4;stroke:#666}#mermaid-svg-vhE0KWMCE0KhqTAy .activation1{fill:#f4f4f4;stroke:#666}#mermaid-svg-vhE0KWMCE0KhqTAy .activation2{fill:#f4f4f4;stroke:#666}#mermaid-svg-vhE0KWMCE0KhqTAy .mermaid-main-font{font-family:"trebuchet ms", verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .section{stroke:none;opacity:0.2}#mermaid-svg-vhE0KWMCE0KhqTAy .section0{fill:rgba(102,102,255,0.49)}#mermaid-svg-vhE0KWMCE0KhqTAy .section2{fill:#fff400}#mermaid-svg-vhE0KWMCE0KhqTAy .section1,#mermaid-svg-vhE0KWMCE0KhqTAy .section3{fill:#fff;opacity:0.2}#mermaid-svg-vhE0KWMCE0KhqTAy .sectionTitle0{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .sectionTitle1{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .sectionTitle2{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .sectionTitle3{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .sectionTitle{text-anchor:start;font-size:11px;text-height:14px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .grid .tick{stroke:#d3d3d3;opacity:0.8;shape-rendering:crispEdges}#mermaid-svg-vhE0KWMCE0KhqTAy .grid .tick text{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .grid path{stroke-width:0}#mermaid-svg-vhE0KWMCE0KhqTAy .today{fill:none;stroke:red;stroke-width:2px}#mermaid-svg-vhE0KWMCE0KhqTAy .task{stroke-width:2}#mermaid-svg-vhE0KWMCE0KhqTAy .taskText{text-anchor:middle;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .taskText:not([font-size]){font-size:11px}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutsideRight{fill:#000;text-anchor:start;font-size:11px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutsideLeft{fill:#000;text-anchor:end;font-size:11px}#mermaid-svg-vhE0KWMCE0KhqTAy .task.clickable{cursor:pointer}#mermaid-svg-vhE0KWMCE0KhqTAy .taskText.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutsideLeft.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutsideRight.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-vhE0KWMCE0KhqTAy .taskText0,#mermaid-svg-vhE0KWMCE0KhqTAy .taskText1,#mermaid-svg-vhE0KWMCE0KhqTAy .taskText2,#mermaid-svg-vhE0KWMCE0KhqTAy .taskText3{fill:#fff}#mermaid-svg-vhE0KWMCE0KhqTAy .task0,#mermaid-svg-vhE0KWMCE0KhqTAy .task1,#mermaid-svg-vhE0KWMCE0KhqTAy .task2,#mermaid-svg-vhE0KWMCE0KhqTAy .task3{fill:#8a90dd;stroke:#534fbc}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutside0,#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutside2{fill:#000}#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutside1,#mermaid-svg-vhE0KWMCE0KhqTAy .taskTextOutside3{fill:#000}#mermaid-svg-vhE0KWMCE0KhqTAy .active0,#mermaid-svg-vhE0KWMCE0KhqTAy .active1,#mermaid-svg-vhE0KWMCE0KhqTAy .active2,#mermaid-svg-vhE0KWMCE0KhqTAy .active3{fill:#bfc7ff;stroke:#534fbc}#mermaid-svg-vhE0KWMCE0KhqTAy .activeText0,#mermaid-svg-vhE0KWMCE0KhqTAy .activeText1,#mermaid-svg-vhE0KWMCE0KhqTAy .activeText2,#mermaid-svg-vhE0KWMCE0KhqTAy .activeText3{fill:#000 !important}#mermaid-svg-vhE0KWMCE0KhqTAy .done0,#mermaid-svg-vhE0KWMCE0KhqTAy .done1,#mermaid-svg-vhE0KWMCE0KhqTAy .done2,#mermaid-svg-vhE0KWMCE0KhqTAy .done3{stroke:grey;fill:#d3d3d3;stroke-width:2}#mermaid-svg-vhE0KWMCE0KhqTAy .doneText0,#mermaid-svg-vhE0KWMCE0KhqTAy .doneText1,#mermaid-svg-vhE0KWMCE0KhqTAy .doneText2,#mermaid-svg-vhE0KWMCE0KhqTAy .doneText3{fill:#000 !important}#mermaid-svg-vhE0KWMCE0KhqTAy .crit0,#mermaid-svg-vhE0KWMCE0KhqTAy .crit1,#mermaid-svg-vhE0KWMCE0KhqTAy .crit2,#mermaid-svg-vhE0KWMCE0KhqTAy .crit3{stroke:#f88;fill:red;stroke-width:2}#mermaid-svg-vhE0KWMCE0KhqTAy .activeCrit0,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCrit1,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCrit2,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCrit3{stroke:#f88;fill:#bfc7ff;stroke-width:2}#mermaid-svg-vhE0KWMCE0KhqTAy .doneCrit0,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCrit1,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCrit2,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCrit3{stroke:#f88;fill:#d3d3d3;stroke-width:2;cursor:pointer;shape-rendering:crispEdges}#mermaid-svg-vhE0KWMCE0KhqTAy .milestone{transform:rotate(45deg) scale(0.8, 0.8)}#mermaid-svg-vhE0KWMCE0KhqTAy .milestoneText{font-style:italic}#mermaid-svg-vhE0KWMCE0KhqTAy .doneCritText0,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCritText1,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCritText2,#mermaid-svg-vhE0KWMCE0KhqTAy .doneCritText3{fill:#000 !important}#mermaid-svg-vhE0KWMCE0KhqTAy .activeCritText0,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCritText1,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCritText2,#mermaid-svg-vhE0KWMCE0KhqTAy .activeCritText3{fill:#000 !important}#mermaid-svg-vhE0KWMCE0KhqTAy .titleText{text-anchor:middle;font-size:18px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy g.classGroup text{fill:#9370db;stroke:none;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:10px}#mermaid-svg-vhE0KWMCE0KhqTAy g.classGroup text .title{font-weight:bolder}#mermaid-svg-vhE0KWMCE0KhqTAy g.clickable{cursor:pointer}#mermaid-svg-vhE0KWMCE0KhqTAy g.classGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-vhE0KWMCE0KhqTAy g.classGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy .classLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.5}#mermaid-svg-vhE0KWMCE0KhqTAy .classLabel .label{fill:#9370db;font-size:10px}#mermaid-svg-vhE0KWMCE0KhqTAy .relation{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-vhE0KWMCE0KhqTAy .dashed-line{stroke-dasharray:3}#mermaid-svg-vhE0KWMCE0KhqTAy #compositionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #compositionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #aggregationStart{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #aggregationEnd{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #dependencyStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #dependencyEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #extensionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy #extensionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy .commit-id,#mermaid-svg-vhE0KWMCE0KhqTAy .commit-msg,#mermaid-svg-vhE0KWMCE0KhqTAy .branch-label{fill:lightgrey;color:lightgrey;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .pieTitleText{text-anchor:middle;font-size:25px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .slice{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy g.stateGroup text{fill:#9370db;stroke:none;font-size:10px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy g.stateGroup text{fill:#9370db;fill:#333;stroke:none;font-size:10px}#mermaid-svg-vhE0KWMCE0KhqTAy g.statediagram-cluster .cluster-label text{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy g.stateGroup .state-title{font-weight:bolder;fill:#000}#mermaid-svg-vhE0KWMCE0KhqTAy g.stateGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-vhE0KWMCE0KhqTAy g.stateGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-vhE0KWMCE0KhqTAy .transition{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-vhE0KWMCE0KhqTAy .stateGroup .composit{fill:white;border-bottom:1px}#mermaid-svg-vhE0KWMCE0KhqTAy .stateGroup .alt-composit{fill:#e0e0e0;border-bottom:1px}#mermaid-svg-vhE0KWMCE0KhqTAy .state-note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-vhE0KWMCE0KhqTAy .state-note text{fill:black;stroke:none;font-size:10px}#mermaid-svg-vhE0KWMCE0KhqTAy .stateLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-vhE0KWMCE0KhqTAy .edgeLabel text{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-vhE0KWMCE0KhqTAy .node circle.state-start{fill:black;stroke:black}#mermaid-svg-vhE0KWMCE0KhqTAy .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-vhE0KWMCE0KhqTAy #statediagram-barbEnd{fill:#9370db}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-state .divider{stroke:#9370db}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-vhE0KWMCE0KhqTAy .note-edge{stroke-dasharray:5}#mermaid-svg-vhE0KWMCE0KhqTAy .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-vhE0KWMCE0KhqTAy .error-icon{fill:#522}#mermaid-svg-vhE0KWMCE0KhqTAy .error-text{fill:#522;stroke:#522}#mermaid-svg-vhE0KWMCE0KhqTAy .edge-thickness-normal{stroke-width:2px}#mermaid-svg-vhE0KWMCE0KhqTAy .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-vhE0KWMCE0KhqTAy .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-vhE0KWMCE0KhqTAy .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-vhE0KWMCE0KhqTAy .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-vhE0KWMCE0KhqTAy .marker{fill:#333}#mermaid-svg-vhE0KWMCE0KhqTAy .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-vhE0KWMCE0KhqTAy {color: rgba(0, 0, 0, 0.75);font: ;}

I0:
S'->.S,#
S->.aAd,#
S->.bAc,#
S->.aec,#
S->.bed,#

I1:
S'->S.,#

I2:
S->a.Ad,#
S->a.ec,#
A->.e,d

4.go(I0,b)
若有项目 S->b.Ac,#属于I3，A→e是文法中的产生式，{c}∈V*，c∈FIRST({c#})，则 A→·e,c 也属于I2中

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text{fill:#9370db;stroke:none;font-size:10px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-oZGvt0rW9HZnb8JD g.stateGroup text{fill:#9370db;fill:#333;stroke:none;font-size:10px}#mermaid-svg-oZGvt0rW9HZnb8JD g.statediagram-cluster .cluster-label text{fill:#333}#mermaid-svg-oZGvt0rW9HZnb8JD g.stateGroup .state-title{font-weight:bolder;fill:#000}#mermaid-svg-oZGvt0rW9HZnb8JD g.stateGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-oZGvt0rW9HZnb8JD g.stateGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-oZGvt0rW9HZnb8JD .transition{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-oZGvt0rW9HZnb8JD .stateGroup .composit{fill:white;border-bottom:1px}#mermaid-svg-oZGvt0rW9HZnb8JD .stateGroup .alt-composit{fill:#e0e0e0;border-bottom:1px}#mermaid-svg-oZGvt0rW9HZnb8JD .state-note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-oZGvt0rW9HZnb8JD .state-note text{fill:black;stroke:none;font-size:10px}#mermaid-svg-oZGvt0rW9HZnb8JD .stateLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-oZGvt0rW9HZnb8JD .edgeLabel text{fill:#333}#mermaid-svg-oZGvt0rW9HZnb8JD .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-oZGvt0rW9HZnb8JD .node circle.state-start{fill:black;stroke:black}#mermaid-svg-oZGvt0rW9HZnb8JD .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-oZGvt0rW9HZnb8JD #statediagram-barbEnd{fill:#9370db}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-state .divider{stroke:#9370db}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-oZGvt0rW9HZnb8JD .note-edge{stroke-dasharray:5}#mermaid-svg-oZGvt0rW9HZnb8JD .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-oZGvt0rW9HZnb8JD .error-icon{fill:#522}#mermaid-svg-oZGvt0rW9HZnb8JD .error-text{fill:#522;stroke:#522}#mermaid-svg-oZGvt0rW9HZnb8JD .edge-thickness-normal{stroke-width:2px}#mermaid-svg-oZGvt0rW9HZnb8JD .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-oZGvt0rW9HZnb8JD .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-oZGvt0rW9HZnb8JD .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-oZGvt0rW9HZnb8JD .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-oZGvt0rW9HZnb8JD .marker{fill:#333}#mermaid-svg-oZGvt0rW9HZnb8JD .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-oZGvt0rW9HZnb8JD {color: rgba(0, 0, 0, 0.75);font: ;}

I0:
S'->.S,#
S->.aAd,#
S->.bAc,#
S->.aec,#
S->.bed,#

I1:
S'->S.,#

I2:
S->a.Ad,#
S->a.ec,#
A->.e,d

I3:
S->b.Ac,#
S->b.ed,#
A->.e,c

5.最终集

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I0:
S'->.S,#
S->.aAd,#
S->.bAc,#
S->.aec,#
S->.bed,#

I1:
S'->S.,#

I2:
S->a.Ad,#
S->a.ec,#
A->.e,d

I3:
S->b.Ac,#
S->b.ed,#
A->.e,c

I4:
S->aA.d,#

I5:
>S->ae.c,#
A->e.,d

I6:
S->bA.c,#

I7:
S->be.d,#
A->e.,c

I8:
S->aAd.,#

I9:
S->ae.c,#

I10:
S->bAc.,#

I11:
S->bed.,#

LR(1)分析表构造

状态	ACTION						GOTO
状态	a	b	c	d	e	#	S	A
0	S2	S3					1
1						acc
2					S5			4
3					S7			6
4				S8
5			S9	r5
6			S10
7			r5	S11
8						r1
9						r3
10						r2
11						r4

LALR(1)分析

LR(1)分析表的构造对搜索符的计算方法比较确切，对文法放宽了要求，可以解决SLR(1)方法中解决不了的问题，介是，由于它的构造对某些同心集的分裂对状态数目引起剧烈增长，从而导致存储容量增加，为了克服LR(1)的这种缺点，我们可采用对LR(1)项目集规范族合并同心集的方法，若合并的同心集不产生冲突，则为LALR(1)项目集

详情(略)

二义性文法在LR分析中的应用

我们已经知道任何一个二义性文法绝不是LR类文法，也不是一个算符优先文法或者LL(K)文法，任何一个二义性文法不存在与其相应的确定的语法分析器，但是对于基些二义性文法，我们可以人为地给出优先性和结合性的规定，从而可以构造出比相应非二义性文法更优越的LR分析器

给定文法G[E’]：
E’->E
E -> E + E
E ->E * E
E -> (E)
E -> i

LR(0)项目集构造

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(

)

I0:
E'->.E
E->.E+E
E->.E*E
E->.(E)
E->.i

I1:
E'->E.
E->E.+E
E->E.*E

I2:
E->(.E)
E->.E+E
E->.E*E
E->.(E)
E->.i

I3:
E->i.

I4:
E->E+.E
E->.E+E
E->.E*E
E->.(E)
E->.i

I5:
E->E*.E
E->.E+E
E->.E*E
E->.(E)
E->.i

I6:
E->(E.)
E->E.+E
E->E.*E

I7:
E->E+E.
E->E.+E
E->E.*E

I8:
E->E*E.
E->E.+E
E->E.*E

I9:
E->(E).

结合follow(E)={+,*,),#}得分析表

状态	ACTION						GOTO
状态	+	*	(	)	i	#	E
0			S2		S3		1
1	S4	S5				acc
2			S2		S3		6
3	r4	r4		r4		r4
4			S2		S3		7
5			S2		S3		8
6	S4	S5		S9
7	r1,S4	r1,S5		r1		r1
8	r2,S4	r2.S5		r2		r2
9	r3	r3		r3		r3

从图中状态可以看出I7和I8中存在移进-归约冲突。但用SLR(1)和LR(K)仍不能解决。然而我们可以用优先关系和结合性可以解决。

1.规定算法的优先关系如下：

1）*优先级最高，具有左结合性
2）+优先级最低，具有左结合性
3）(和)，括号的优先级小于括号内运算符，大于括号外运算符优先级；内括号的优先级大于外括号的优先级
4）任何于#相邻的符号的优先级都大于#

2.算符优先关系表如下

	+	*	(	)	i	#
+	>	<	<	>	<	>
*	>	>	<	>	<	>
(	<	<	<		<
)	>	>		>		>
)	>	>		>		>
#	<	<	<		<	=

3.结合算符优先关系表改造得

在i7中，前算符为+，
1.因算符优先关系表中[+>+]进行归约
2.因算符优先关系表中[+<]进行移进
在i8中，前算符为
1.因算符优先关系表中[>+]进行归约
2.因算符优先关系表中[>*]进行归约

状态	ACTION						GOTO
状态	+	*	(	)	i	#	E
0			S2		S3		1
1	S4	S5				acc
2			S2		S3		6
3	r4	r4		r4		r4
4			S2		S3		7
5			S2		S3		8
6	S4	S5		S9
7	r1	S5		r1		r1
8	r2	r2		r2		r2
9	r3	r3		r3		r3

4.输入字符串i + i * i

步骤	状态栈	符号栈	输入符串	动作
1	0	#	i+i*i#	移进S2
2	03	#i	+i*i#	归约r4转到【1】
3	01	#E	+i*i#	移进S4
4	014	#E+	i*i#	移进S3
5	0143	#E+i	*i#	归约r4转到【7】
6	0147	#E+E	* i#	移进S5
7	01475	#E+E*	i#	移进S3
8	014753	#E+E*i	#	归约r4转到【8】
9	014758	#E+E*E	#	归约r2转到【7】
10	0147	#E+E	#	归约r1转到【1】
11	01	#E	#	acc