编译原理(五) LL(1)文法分析法(预测分析表的构造算法C++实现)

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 507using 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epsilonclass 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();    }}
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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 507using 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epsilonclass 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();    }}
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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 507using 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epsilonclass 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();    }}
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