## 基本定义
- FIRST(α):
- 令Ｇ是一个不含左递归的文法，对G的所有非终结符的每个候选α定义它的终结首符集FIRST(α)为：
FIRST(α)={a | α=>*a…, a∈VT}
- 若α=>*ε，则规定ε∈FIRST(α)
- FIRST(α)是α的所有可能推导的开头终结符或可能的ε
- 如果非终结符Ａ的所有候选首符集两两不相交，即Ａ的任何两个不同候选αi和αｊ
FIRST(αi) ∩FIRST(αj)=Φ
- 那么当要求Ａ匹配输入串时，Ａ就能根据它所面临的第一个输入符号ａ，准确的指派某一个候选前去执行任务。这个候选就是那个终结首符集含ａ的α。
- FOLLOW(A):
- 假定S是文法G的开始符号，对于G的任何非终结符A，我们定义
FOLLOW(A)={a | S=>*…Aa…，a∈VT}
- FOLLOW(A)是所有句型中出现在紧接A之后的终结符或“#”。开始符号的FOLLOW集初始化时加入“#”。
- 当非终结符A面临输入符号a，且a不属于A的任意候选首符集但A的某个候选首符集包含ε时，只有当a∈FOLLOW(A)时，才可能允许A自动匹配。
- LL(1)文法:
- 文法不含左递归
- 对于文法中每一个非终结符A的各个产生式的候选首符集两两不相交。即，若A→α1 |α2 | … |αn，则FIRST(αi)∩FIRST(αj)=Φ (i≠j)
- 对文法中的每个非终结符A，若它存在某个候选首符集包含ε，则，FIRST(A)∩FOLLOW(A)=Φ
如果一个文法G满足以上条件，则称该文法G为***LL(1)文法***。
- LL(1)文法是***不带回溯***的***自顶向下***的文法
- 预测分析表：
- 预测分析表示一个M[A,a]形式的矩阵。其中A为非终结符，a是终结符或‘#’ 。
- 矩阵元素M[A,a]中存放着一条关于A的产生式，指出当A面临输入符号a时所应采用的候选。
M[A,a]中也可能存放一个“出错标志”，指出A根本不该面临输入符号a。
## 预测分析表的构造
### FIRST(α)的构造算法
要构造FIRST(α)，根据定义：
α=X1⋯Xn
那么对于从前到后的Xi我们进行分类讨论：
- 如果Xi∈Vt,那么FIRST(α)=FIRST(Xi)={Xi}
- 如果Xi∈Vn,因为不存在左递归，所以Xi=a.......|ϵ,那么FIRST(Xi)={a,ϵ,FIRST(Xi+1)}
- 只要Xi−1不包含ϵ,那么Xi不可能影响FIRST(α)
- 那么我们通过记录每个a∈V,然后进行深度优先记忆化搜索,将所有的状态填满，因为LL(1)文法使不会回溯的，所以能够保证在O(n)的时间完成，采取递归的形式实现
### FIRST(α)的构造实现代码
~~~
#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <cstdlib>
#include <vector>
#include <string>
#include <cctype>
#include <map>
#include <set>
#define MAX 507
using namespace std;
//大写字母为非终止符(可以多一个'的标号做区分)，小写字母为终止符,用~代替epsilon
class WF
{
public:
string left;
set<string> right;
WF ( char s[] )
{
left = s;
}
void print ( )
{
printf ( "%s->" , left.c_str() );
set<string>::iterator it = right.begin();
if ( right.begin()!= right.end() )
{
printf ( "%s" , it->c_str() );
it++;
}
for(; it != right.end() ; it++ )
printf ( "|%s" , it->c_str() );
puts("");
}
void insert ( char s[] )
{
right.insert ( s );
}
};
map<string,set<char> > first;
map<string,set<char> > follow;
map<string,int> VN_dic;
vector<WF> VN_set;
bool used[MAX];
void dfs ( int x )
{
if ( used[x] ) return;
used[x] = 1;
string& left = VN_set[x].left;
set<string>& right = VN_set[x].right;
set<string>::iterator it = right.begin();
for ( ; it!= right.end() ; it++ )
for ( int i = 0 ; i < it->length() ; i++ )
{
if ( !isupper( it->at(i) ) && it->at(i) != '\'' )
{
first[left].insert ( it->at(i) );
break;
}
if ( isupper( it->at(i) ) )
{
int y;
if ( i != it->length()-1 && it->at(i+1) == '\'' )
y = VN_dic[it->substr(i,2)]-1;
else y = VN_dic[it->substr(i,1)]-1;
string& tleft = VN_set[y].left;
dfs ( y );
set<char>& temp = first[tleft];
set<char>::iterator it1 = temp.begin();
bool flag = true;
for ( ; it1 != temp.end() ; it1++ )
{
if ( *it1 == '~' ) flag = false;
first[left].insert( *it1 );
}
if ( flag ) break;
}
else continue;
}
}
void make_first ( )
{
memset ( used , 0 , sizeof ( used ) );
for ( int i = 0 ; i < VN_set.size() ; i++ )
dfs ( i );
#define DEBUG
#ifdef DEBUG
map<string,set<char> >::iterator it = first.begin();
for ( ; it != first.end() ; it++ )
{
printf ( "FIRST(%s)={" , it->first.c_str() );
set<char> & temp = it->second;
set<char>::iterator it1 = temp.begin();
bool flag = false;
for ( ; it1 != temp.end() ; it1++ )
{
if ( flag ) printf ( "," );
printf ( "%c" , *it1 );
flag = true;
}
puts ("}");
}
#endif
}
int main ( )
{
int n;
char s[MAX];
while ( ~scanf ( "%d" , &n ) )
{
for ( int i = 0 ; i < n ; i++ )
{
scanf ( "%s" , s );
int len = strlen ( s ),j;
for ( j = 0 ; j < len ; j++ )
if ( s[j] == '-' ) break;
s[j] = 0;
if ( !VN_dic[s] )
{
VN_set.push_back ( s );
VN_dic[s] = VN_set.size();
}
int x = VN_dic[s]-1;
VN_set[x].insert ( s+j+2 );
}
make_first();
}
}
~~~
**Input:**
**Output:**
### FOLLOW(A)的构造算法
设S,A,B∈Vn,那么连续使用如下规则，直至follow集不再发生变化：
- （1） S为标识符，那么FOLLOW(S)包含“#”
- （2） 若A::=αBβ,那么FOLLOW(B)+=FIRST(B)−{ϵ}
- （3） 若A::=αB或者A::=αBβ,且β⇒∗ϵ，那么FOLLOW(B)+=FOLLOW(A)
### FOLLOW(A)的构造实现代码
~~~
#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <cstdlib>
#include <vector>
#include <string>
#include <cctype>
#include <map>
#include <set>
#define MAX 507
using namespace std;
//大写字母为非终止符(可以多一个'的标号做区分)，小写字母为终止符,用~代替epsilon
class WF
{
public:
string left;
set<string> right;
WF ( char s[] )
{
left = s;
}
void print ( )
{
printf ( "%s->" , left.c_str() );
set<string>::iterator it = right.begin();
if ( right.begin()!= right.end() )
{
printf ( "%s" , it->c_str() );
it++;
}
for(; it != right.end() ; it++ )
printf ( "|%s" , it->c_str() );
puts("");
}
void insert ( char s[] )
{
right.insert ( s );
}
};
map<string,set<char> > first;
map<string,set<char> > follow;
map<string,int> VN_dic;
vector<WF> VN_set;
bool used[MAX];
void dfs ( int x )
{
if ( used[x] ) return;
used[x] = 1;
string& left = VN_set[x].left;
set<string>& right = VN_set[x].right;
set<string>::iterator it = right.begin();
for ( ; it!= right.end() ; it++ )
for ( int i = 0 ; i < it->length() ; i++ )
{
if ( !isupper( it->at(i) ) && it->at(i) != '\'' )
{
first[left].insert ( it->at(i) );
break;
}
if ( isupper( it->at(i) ) )
{
int y;
if ( i != it->length()-1 && it->at(i+1) == '\'' )
y = VN_dic[it->substr(i,2)]-1;
else y = VN_dic[it->substr(i,1)]-1;
string& tleft = VN_set[y].left;
dfs ( y );
set<char>& temp = first[tleft];
set<char>::iterator it1 = temp.begin();
bool flag = true;
for ( ; it1 != temp.end() ; it1++ )
{
if ( *it1 == '~' ) flag = false;
first[left].insert( *it1 );
}
if ( flag ) break;
}
else continue;
}
}
void make_first ( )
{
memset ( used , 0 , sizeof ( used ) );
for ( int i = 0 ; i < VN_set.size() ; i++ )
dfs ( i );
#define DEBUG
#ifdef DEBUG
puts ("***************FIRST集***********************");
map<string,set<char> >::iterator it = first.begin();
for ( ; it != first.end() ; it++ )
{
printf ( "FIRST(%s)={" , it->first.c_str() );
set<char> & temp = it->second;
set<char>::iterator it1 = temp.begin();
bool flag = false;
for ( ; it1 != temp.end() ; it1++ )
{
if ( flag ) printf ( "," );
printf ( "%c" , *it1 );
flag = true;
}
puts ("}");
}
#endif
}
void append ( const string& str1 , const string& str2 )
{
set<char>& from = follow[str1];
set<char>& to = follow[str2];
set<char>::iterator it = from.begin();
for ( ; it != from.end() ; it++ )
to.insert ( *it );
}
void make_follow ( )
{
while ( true )
{
bool goon = false;
for ( int i = 0 ; i < VN_set.size() ; i++ )
{
string& left = VN_set[i].left;
set<string>& right = VN_set[i].right;
set<string>::iterator it = right.begin();
for ( ; it!= right.end() ; it++ )
{
bool flag = true;
const string& str = *it;
for ( int j = it->length()-1 ; j >= 0 ; j-- )
{
if ( str[j] == '\'' )
{
int x = VN_dic[it->substr(j-1,2)]-1;
if ( flag )
{
int tt = follow[it->substr(j-1,2)].size();
append ( left , it->substr(j-1,2) );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[x].right.count("~" ) )
flag = false;
}
for ( int k = j+1 ; k < it->length() ; k++ )
{
if ( isupper(str[k]) )
{
string id;
if ( k != it->length()-1 && str[k+1] == '\'' )
id = it->substr(k,2);
else id = it->substr(k,1);
set<char>& from = first[id];
set<char>& to = follow[it->substr(j-1,2)];
int tt = to.size();
set<char>::iterator it1 = from.begin();
for ( ; it1 != from.end() ; it1++ )
if ( *it1 != '~' )
to.insert ( *it1 );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[VN_dic[id]-1].right.count("~") )
break;
}
else if ( str[k] != '\'' )
{
int tt = follow[it->substr(j-1,2)].size();
follow[it->substr(j-1,2)].insert ( str[k] );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt )
goon = true;
break;
}
else continue;
}
j--;
}
else if ( isupper(str[j] ) )
{
int x = VN_dic[it->substr(j,1)]-1;
if ( flag )
{
int tt = follow[it->substr(j,1)].size();
append ( left , it->substr(j,1) );
if ( !VN_set[x].right.count("~") )
flag = false;
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
}
for ( int k = j+1 ; k < it->length() ; k++ )
{
if ( isupper( str[k] ) )
{
string id;
if ( k != it->length()-1 && str[k+1] == '\'' )
id = it->substr(k,2);
else id = it->substr(k,1);
set<char>& from = first[id];
set<char>& to = follow[it->substr(j,1)];
set<char>::iterator it1 = from.begin();
int tt = follow[it->substr(j,1)].size();
for ( ; it1 != from.end() ; it1++ )
if ( *it1 != '~' )
to.insert( *it1 );
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[VN_dic[id]-1].right.count("~") )
break;
}
else if ( str[k] != '\'' )
{
int tt = follow[it->substr(j,1)].size();
follow[it->substr(j,1)].insert ( str[k] );
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
break;
}
else continue;
}
}
else flag = false;
}
}
}
if ( !goon ) break;
}
#define DEBUG
#ifdef DEBUG
puts ("****************FOLLOW集**********************" );
map<string,set<char> >::iterator it = follow.begin();
for ( ; it != follow.end() ; it++ )
{
printf ( "FOLLOW(%s)={" , it->first.c_str() );
set<char> & temp = it->second;
temp.insert('#');
set<char>::iterator it1 = temp.begin();
bool flag = false;
for ( ; it1 != temp.end() ; it1++ )
{
if ( flag ) printf ( "," );
printf ( "%c" , *it1 );
flag = true;
}
puts ("}");
}
#endif
}
int main ( )
{
int n;
char s[MAX];
while ( ~scanf ( "%d" , &n ) )
{
for ( int i = 0 ; i < n ; i++ )
{
scanf ( "%s" , s );
int len = strlen ( s ),j;
for ( j = 0 ; j < len ; j++ )
if ( s[j] == '-' ) break;
s[j] = 0;
if ( !VN_dic[s] )
{
VN_set.push_back ( s );
VN_dic[s] = VN_set.size();
}
int x = VN_dic[s]-1;
VN_set[x].insert ( s+j+2 );
}
make_first();
make_follow();
}
}
~~~
**Input:**
**Output:**
### 预测分析表的构造算法
### 预测分析表的构造实现代码
~~~
#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <cstdlib>
#include <vector>
#include <string>
#include <cctype>
#include <map>
#include <set>
#define MAX 507
using namespace std;
//大写字母为非终止符(可以多一个'的标号做区分)，小写字母为终止符,用~代替epsilon
class WF
{
public:
string left;
set<string> right;
WF ( char s[] )
{
left = s;
}
void print ( )
{
printf ( "%s->" , left.c_str() );
set<string>::iterator it = right.begin();
if ( right.begin()!= right.end() )
{
printf ( "%s" , it->c_str() );
it++;
}
for(; it != right.end() ; it++ )
printf ( "|%s" , it->c_str() );
puts("");
}
void insert ( char s[] )
{
right.insert ( s );
}
};
map<string,set<char> > first;
map<string,set<char> > follow;
map<string,int> VN_dic;
vector<WF> VN_set;
bool used[MAX];
void dfs ( int x )
{
if ( used[x] ) return;
used[x] = 1;
string& left = VN_set[x].left;
set<string>& right = VN_set[x].right;
set<string>::iterator it = right.begin();
for ( ; it!= right.end() ; it++ )
for ( int i = 0 ; i < it->length() ; i++ )
{
if ( !isupper( it->at(i) ) && it->at(i) != '\'' )
{
first[left].insert ( it->at(i) );
break;
}
if ( isupper( it->at(i) ) )
{
int y;
if ( i != it->length()-1 && it->at(i+1) == '\'' )
y = VN_dic[it->substr(i,2)]-1;
else y = VN_dic[it->substr(i,1)]-1;
string& tleft = VN_set[y].left;
dfs ( y );
set<char>& temp = first[tleft];
set<char>::iterator it1 = temp.begin();
bool flag = true;
for ( ; it1 != temp.end() ; it1++ )
{
if ( *it1 == '~' ) flag = false;
first[left].insert( *it1 );
}
if ( flag ) break;
}
else continue;
}
}
void make_first ( )
{
memset ( used , 0 , sizeof ( used ) );
for ( int i = 0 ; i < VN_set.size() ; i++ )
dfs ( i );
#define DEBUG
#ifdef DEBUG
puts ("***************FIRST集***********************");
map<string,set<char> >::iterator it = first.begin();
for ( ; it != first.end() ; it++ )
{
printf ( "FIRST(%s)={" , it->first.c_str() );
set<char> & temp = it->second;
set<char>::iterator it1 = temp.begin();
bool flag = false;
for ( ; it1 != temp.end() ; it1++ )
{
if ( flag ) printf ( "," );
printf ( "%c" , *it1 );
flag = true;
}
puts ("}");
}
#endif
}
void append ( const string& str1 , const string& str2 )
{
set<char>& from = follow[str1];
set<char>& to = follow[str2];
set<char>::iterator it = from.begin();
for ( ; it != from.end() ; it++ )
to.insert ( *it );
}
void make_follow ( )
{
while ( true )
{
bool goon = false;
for ( int i = 0 ; i < VN_set.size() ; i++ )
{
string& left = VN_set[i].left;
set<string>& right = VN_set[i].right;
set<string>::iterator it = right.begin();
for ( ; it!= right.end() ; it++ )
{
bool flag = true;
const string& str = *it;
for ( int j = it->length()-1 ; j >= 0 ; j-- )
{
if ( str[j] == '\'' )
{
int x = VN_dic[it->substr(j-1,2)]-1;
if ( flag )
{
int tt = follow[it->substr(j-1,2)].size();
append ( left , it->substr(j-1,2) );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[x].right.count("~" ) )
flag = false;
}
for ( int k = j+1 ; k < it->length() ; k++ )
{
if ( isupper(str[k]) )
{
string id;
if ( k != it->length()-1 && str[k+1] == '\'' )
id = it->substr(k,2);
else id = it->substr(k,1);
set<char>& from = first[id];
set<char>& to = follow[it->substr(j-1,2)];
int tt = to.size();
set<char>::iterator it1 = from.begin();
for ( ; it1 != from.end() ; it1++ )
if ( *it1 != '~' )
to.insert ( *it1 );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[VN_dic[id]-1].right.count("~") )
break;
}
else if ( str[k] != '\'' )
{
int tt = follow[it->substr(j-1,2)].size();
follow[it->substr(j-1,2)].insert ( str[k] );
int tt1 = follow[it->substr(j-1,2)].size();
if ( tt1 > tt )
goon = true;
break;
}
else continue;
}
j--;
}
else if ( isupper(str[j] ) )
{
int x = VN_dic[it->substr(j,1)]-1;
if ( flag )
{
int tt = follow[it->substr(j,1)].size();
append ( left , it->substr(j,1) );
if ( !VN_set[x].right.count("~") )
flag = false;
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
}
for ( int k = j+1 ; k < it->length() ; k++ )
{
if ( isupper( str[k] ) )
{
string id;
if ( k != it->length()-1 && str[k+1] == '\'' )
id = it->substr(k,2);
else id = it->substr(k,1);
set<char>& from = first[id];
set<char>& to = follow[it->substr(j,1)];
set<char>::iterator it1 = from.begin();
int tt = follow[it->substr(j,1)].size();
for ( ; it1 != from.end() ; it1++ )
if ( *it1 != '~' )
to.insert( *it1 );
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
if ( !VN_set[VN_dic[id]-1].right.count("~") )
break;
}
else if ( str[k] != '\'' )
{
int tt = follow[it->substr(j,1)].size();
follow[it->substr(j,1)].insert ( str[k] );
int tt1 = follow[it->substr(j,1)].size();
if ( tt1 > tt ) goon = true;
break;
}
else continue;
}
}
else flag = false;
}
}
}
if ( !goon ) break;
}
#define DEBUG
#ifdef DEBUG
puts ("****************FOLLOW集**********************" );
map<string,set<char> >::iterator it = follow.begin();
for ( ; it != follow.end() ; it++ )
{
printf ( "FOLLOW(%s)={" , it->first.c_str() );
set<char> & temp = it->second;
temp.insert('#');
set<char>::iterator it1 = temp.begin();
bool flag = false;
for ( ; it1 != temp.end() ; it1++ )
{
if ( flag ) printf ( "," );
printf ( "%c" , *it1 );
flag = true;
}
puts ("}");
}
#endif
}
vector<map<char,string> > predict_table;
//检查一个字符是否属于一个字符串的FIRST集合
bool check_first ( const string& text , char ch )
{
for ( int i = 0 ; i < text.length() ; i++ )
{
bool hasEmpty = false;
if ( !isupper(text[i]) && text[i] != '\'' )
{
if ( text[i] != ch ) return false;
else return true;
}
else if ( isupper(text[i] ) )
{
string temp;
if ( i != text.length()-1 && text[i+1] == '\'' )
temp = text.substr(i,2);
else
temp = text.substr(i,1);
set<char>& dic = first[temp];
set<char>::iterator it = dic.begin();
for ( ; it != dic.end() ; it++ )
{
if ( *it == '~' ) hasEmpty = true;
if ( *it == ch ) return true;
}
if ( !hasEmpty) break;
}
else continue;
}
return false;
}
//检查一个字符是否属于一个字符串的FOLLOW集合
bool check_follow ( const string& text , char ch )
{
set<char>& dic = follow[text];
set<char>::iterator it = dic.begin();
for ( ; it != dic.end() ; it++ )
if ( *it == ch ) return true;
return false;
}
void make_table ()
{
map<char,string> temp;
vector<char> letter;
bool vis[500];
memset ( vis , 0 , sizeof ( vis ) );
for ( int i = 0 ; i < VN_set.size() ; i++ )
{
set<string>& right = VN_set[i].right;
set<string>::iterator it = right.begin();
for ( ; it != right.end() ; it++ )
for ( int j = 0 ; j < it->length() ; j++ )
if ( !isupper(it->at(j)) && it->at(j) != '\'' )
{
if ( vis[it->at(j)] ) continue;
vis[it->at(j)] = true;
letter.push_back ( it->at(j) );
}
}
for ( int i = 0 ; i < VN_set.size() ; i++ )
{
temp.clear();
string& left = VN_set[i].left;
set<string>& right = VN_set[i].right;
set<string>::iterator it = right.begin();
for ( ; it != right.end() ; it++ )
for ( int j = 0 ; j < letter.size() ; j++ )
{
//cout << *it << " " << letter[j] << endl;
if ( check_first ( *it , letter[j] ) )
{
//cout << "YES" << endl;
temp[letter[j]] = *it;
}
if ( it->at(0) == '~' && check_follow ( left, letter[j] ))
temp[letter[j]] = *it;
}
predict_table.push_back ( temp );
}
#define DEBUG
#ifdef DEBUG
for ( int i = 0 ; i <= (letter.size()+1)*10 ; i++ )
printf ( "-" );
puts ("");
printf ( "|%9s" , "|" );
for ( int i = 0 ; i < letter.size() ; i++ )
printf ( "%5c%5s" , letter[i] , "|" );
puts("");
for ( int i = 0 ; i <= (letter.size()+1)*10 ; i++ )
printf ( "-" );
puts("");
for ( int i = 0 ; i < VN_set.size() ; i++ )
{
printf ( "|%5s%4s" , VN_set[i].left.c_str() , "|" );
for ( int j = 0 ; j < letter.size() ; j ++ )
if ( predict_table[i].count(letter[j] ) )
printf ( "%7s%3s" , predict_table[i][letter[j]].c_str() , "|" );
else printf ( "%10s" , "|" );
puts("");
for ( int i = 0 ; i <= (letter.size()+1)*10 ; i++ )
printf ( "-" );
puts ("");
}
#endif
}
int main ( )
{
int n;
char s[MAX];
while ( ~scanf ( "%d" , &n ) )
{
for ( int i = 0 ; i < n ; i++ )
{
scanf ( "%s" , s );
int len = strlen ( s ),j;
for ( j = 0 ; j < len ; j++ )
if ( s[j] == '-' ) break;
s[j] = 0;
if ( !VN_dic[s] )
{
VN_set.push_back ( s );
VN_dic[s] = VN_set.size();
}
int x = VN_dic[s]-1;
VN_set[x].insert ( s+j+2 );
}
make_first();
make_follow();
make_table();
}
}
~~~
**Input:**
**Output:**
