用C++设计简单算数表达式语法分析器算法；（LR来实现）输入表达式不带#号，输出为TRue或者False

LR(1) 算法可以用于实现简单算数表达式语法分析器。下面是一个用 C++ 设计的算法实现：

#include <iostream>
#include <string>
#include <stack>
#include <map>
#include <vector>

using namespace std;

const int MAXN = 1000;

// 符号类别
enum TokenType {
    TOKEN_PLUS,     // +
    TOKEN_MINUS,    // -
    TOKEN_MULTIPLY, // *
    TOKEN_DIVIDE,   // /
    TOKEN_LPAREN,   // (
    TOKEN_RPAREN,   // )
    TOKEN_NUMBER    // 数字
};

// 符号项
struct Token {
    TokenType type; // 符号类别
    int value;      // 数字值
};

// LR(1) 项
struct LR1Item {
    int production;     // 产生式编号
    int dot;            // 点的位置
    vector<int> lookahead;  // 向前看符号
};

// 产生式
struct Production {
    TokenType left;     // 左部
    vector<TokenType> right;    // 右部
};

// LR(1) 项目集族
vector<vector<LR1Item>> item_sets;

// ACTION 表
map<pair<int, TokenType>, pair<char, int>> action;

// GOTO 表
map<pair<int, TokenType>, int> goto_table;

// 产生式序列
vector<Production> productions = {
    {TOKEN_NUMBER, {TOKEN_NUMBER}},
    {TOKEN_NUMBER, {TOKEN_NUMBER, TOKEN_PLUS, TOKEN_NUMBER}},
    {TOKEN_NUMBER, {TOKEN_NUMBER, TOKEN_MINUS, TOKEN_NUMBER}},
    {TOKEN_NUMBER, {TOKEN_NUMBER, TOKEN_MULTIPLY, TOKEN_NUMBER}},
    {TOKEN_NUMBER, {TOKEN_NUMBER, TOKEN_DIVIDE, TOKEN_NUMBER}},
    {TOKEN_NUMBER, {TOKEN_LPAREN, TOKEN_NUMBER, TOKEN_RPAREN}}
};

// 符号优先级表
int priority_table[MAXN][MAXN] = {
    // +   -   *   /   (   )   number
    { 1,  1, -1, -1, -1,  1, -1}, // +
    { 1,  1, -1, -1, -1,  1, -1}, // -
    { 1,  1,  1,  1, -1,  1, -1}, // *
    { 1,  1,  1,  1, -1,  1, -1}, // /
    {-1, -1, -1, -1, -1,  0, -1}, // (
    { 1,  1,  1,  1, -1,  1, -1}, // )
    {-1, -1, -1, -1, -1, -1,  0}  // number
};

// 将符号转化为字符串
string tokenToString(TokenType type) {
    switch (type) {
    case TOKEN_PLUS:
        return "+";
    case TOKEN_MINUS:
        return "-";
    case TOKEN_MULTIPLY:
        return "*";
    case TOKEN_DIVIDE:
        return "/";
    case TOKEN_LPAREN:
        return "(";
    case TOKEN_RPAREN:
        return ")";
    case TOKEN_NUMBER:
        return "number";
    default:
        return "";
    }
}

// 判断一个符号是否为终结符
bool isTerminal(TokenType type) {
    return type == TOKEN_PLUS || type == TOKEN_MINUS || type == TOKEN_MULTIPLY || type == TOKEN_DIVIDE
        || type == TOKEN_LPAREN || type == TOKEN_RPAREN || type == TOKEN_NUMBER;
}

// 计算表达式的值
int calculate(vector<Token> tokens) {
    stack<int> value_stack;
    stack<TokenType> op_stack;

    for (size_t i = 0; i < tokens.size(); i++) {
        Token token = tokens[i];
        if (token.type == TOKEN_NUMBER) {
            value_stack.push(token.value);
        } else {
            while (!op_stack.empty() && priority_table[op_stack.top()][token.type] >= 0) {
                TokenType op = op_stack.top();
                op_stack.pop();
                int b = value_stack.top();
                value_stack.pop();
                int a = value_stack.top();
                value_stack.pop();
                switch (op) {
                case TOKEN_PLUS:
                    value_stack.push(a + b);
                    break;
                case TOKEN_MINUS:
                    value_stack.push(a - b);
                    break;
                case TOKEN_MULTIPLY:
                    value_stack.push(a * b);
                    break;
                case TOKEN_DIVIDE:
                    value_stack.push(a / b);
                    break;
                }
            }

            if (token.type == TOKEN_RPAREN) {
                op_stack.pop();
            } else {
                op_stack.push(token.type);
            }
        }
    }

    while (!op_stack.empty()) {
        TokenType op = op_stack.top();
        op_stack.pop();
        int b = value_stack.top();
        value_stack.pop();
        int a = value_stack.top();
        value_stack.pop();
        switch (op) {
        case TOKEN_PLUS:
            value_stack.push(a + b);
            break;
        case TOKEN_MINUS:
            value_stack.push(a - b);
            break;
        case TOKEN_MULTIPLY:
            value_stack.push(a * b);
            break;
        case TOKEN_DIVIDE:
            value_stack.push(a / b);
            break;
        }
    }

    return value_stack.top();
}

// 获取产生式字符串
string productionToString(Production production) {
    string result = tokenToString(production.left) + " ->";
    for (size_t i = 0; i < production.right.size(); i++) {
        result += " " + tokenToString(production.right[i]);
    }
    return result;
}

// 获取 LR(1) 项目字符串
string lr1ItemToString(LR1Item item) {
    string result = productionToString(productions[item.production]) + ", dot = " + to_string(item.dot);
    result += ", lookahead = { ";
    for (size_t i = 0; i < item.lookahead.size(); i++) {
        result += tokenToString(item.lookahead[i]) + " ";
    }
    result += "}";
    return result;
}

// 获取项目集字符串
string itemSetToString(int index) {
    string result = "{\n";
    for (size_t i = 0; i < item_sets[index].size(); i++) {
        result += "  " + lr1ItemToString(item_sets[index][i]) + "\n";
    }
    result += "}";
    return result;
}

// 获取 ACTION 表字符串
string actionTableToString() {
    string result = "ACTION TABLE:\n";
    for (auto it = action.begin(); it != action.end(); it++) {
        int state = it->first.first;
        TokenType token = it->first.second;
        char action_type = it->second.first;
        int action_param = it->second.second;
        result += "  [" + to_string(state) + ", " + tokenToString(token) + "] = ";
        if (action_type == 's') {
            result += "SHIFT " + to_string(action_param) + "\n";
        } else if (action_type == 'r') {
            result += "REDUCE " + productionToString(productions[action_param]) + "\n";
        } else if (action_type == 'a') {
            result += "ACCEPT\n";
        }
    }
    return result;
}

// 获取 GOTO 表字符串
string gotoTableToString() {
    string result = "GOTO TABLE:\n";
    for (auto it = goto_table.begin(); it != goto_table.end(); it++) {
        int state = it->first.first;
        TokenType symbol = it->first.second;
        int next_state = it->second;
        result += "  [" + to_string(state) + ", " + tokenToString(symbol) + "] = " + to_string(next_state) + "\n";
    }
    return result;
}

// 初始化符号项
void initItem(Token token, vector<int>& lookahead, vector<LR1Item>& items) {
    for (size_t i = 0; i < productions.size(); i++) {
        Production production = productions[i];
        if (production.left == token.type) {
            LR1Item item;
            item.production = i;
            item.dot = 0;
            item.lookahead = lookahead;
            items.push_back(item);
        }
    }
}

// 计算闭包
void closure(vector<LR1Item>& items) {
    bool changed = true;
    while (changed) {
        changed = false;
        for (size_t i = 0; i < items.size(); i++) {
            LR1Item item = items[i];
            if (item.dot < productions[item.production].right.size() && !isTerminal(productions[item.production].right[item.dot])) {
                TokenType symbol = productions[item.production].right[item.dot];
                vector<int> lookahead = item.lookahead;
                if (item.dot + 1 < productions[item.production].right.size()) {
                    lookahead.clear();
                    lookahead.push_back(productions[item.production].right[item.dot + 1]);
                }
                vector<LR1Item> new_items;
                initItem({symbol, 0}, lookahead, new_items);
                for (size_t j = 0; j < new_items.size(); j++) {
                    if (find(items.begin(), items.end(), new_items[j]) == items.end()) {
                        items.push_back(new_items[j]);
                        changed = true;
                    }
                }
            }
        }
    }
}

// 计算转移
void gotoState(vector<LR1Item> items, TokenType symbol, vector<LR1Item>& new_items) {
    for (size_t i = 0; i < items.size(); i++) {
        LR1Item item = items[i];
        if (item.dot < productions[item.production].right.size() && productions[item.production].right[item.dot] == symbol) {
            LR1Item new_item = item;
            new_item.dot++;
            new_items.push_back(new_item);
        }
    }
    closure(new_items);
}

// 计算项目集族
void calculateItemSets() {
    vector<LR1Item> initial_items;
    initItem({TOKEN_NUMBER, 0}, {TOKEN_NUMBER}, initial_items);
    closure(initial_items);
    item_sets.push_back(initial_items);

    bool changed = true;
    while (changed) {
        changed = false;
        for (size_t i = 0; i < item_sets.size(); i++) {
            vector<LR1Item> items = item_sets[i];
            map<TokenType, vector<LR1Item>> next_items;
            for (size_t j = 0; j < items.size(); j++) {
                LR1Item item = items[j];
                if (item.dot < productions[item.production].right.size()) {
                    TokenType symbol = productions[item.production].right[item.dot];
                    vector<LR1Item> new_items;
                    gotoState(items, symbol, new_items);
                    next_items[symbol] = new_items;
                }
            }
            for (auto it = next_items.begin(); it != next_items.end(); it++) {
                TokenType symbol = it->first;
                vector<LR1Item> new_items = it->second;
                if (new_items.size() > 0 && find(item_sets.begin(), item_sets.end(), new_items) == item_sets.end()) {
                    item_sets.push_back(new_items);
                    changed = true;
                    goto_table[{i, symbol}] = item_sets.size() - 1;
                }
            }
        }
    }
}

// 计算 ACTION 表和 GOTO 表
void calculateTables() {
    for (size_t i = 0; i < item_sets.size(); i++) {
        vector<LR1Item> items = item_sets[i];
        for (size_t j = 0; j < items.size(); j++) {
            LR1Item item = items[j];
            if (item.dot == productions[item.production].right.size()) {
                if (item.production == 0) {
                    action[{i, TOKEN_NUMBER}] = {'a', 0};
                } else {
                    for (size_t k = 0; k < item.lookahead.size(); k++) {
                        TokenType lookahead = item.lookahead[k];
                        if (action.count({i, lookahead}) > 0) {
                            cout << "ERROR: Reduce-Reduce conflict!\n";
                            return;
                        } else {
                            action[{i, lookahead}] = {'r', item.production};
                        }
                    }
                }
            }
        }
        for (auto it = productions.begin(); it != productions.end(); it++) {
            TokenType symbol = it->left;
            vector<LR1Item> new_items;
            gotoState(items, symbol, new_items);
            if (new_items.size() > 0) {
                int j = find(item_sets.begin(), item_sets.end(), new_items) - item_sets.begin();
                goto_table[{i, symbol}] = j;
            }
        }
        for (size_t j = 0; j < items.size(); j++) {
            LR1Item item = items[j];
            if (item.dot < productions[item.production].right.size()) {
                TokenType symbol = productions[item.production].right[item.dot];
                if (isTerminal(symbol)) {
                    vector<LR1Item> new_items;
                    gotoState(items, symbol, new_items);
                    if (new_items.size() > 0) {
                        int j = find(item_sets.begin(), item_sets.end(), new_items) - item_sets.begin();
                        if (action.count({i, symbol}) > 0) {
                            cout << "ERROR: Shift-Reduce conflict!\n";
                            return;
                        } else {
                            action[{i, symbol}] = {'s', j};
                        }
                    }
                }
            }
        }
    }
}

// 将表达式转化为符号串
vector<Token> expressionToTokens(string expr) {
    vector<Token> tokens;
    int i = 0;
    while (i < expr.length()) {
        if (isdigit(expr[i])) {
            int j = i;
            while (j < expr.length() && isdigit(expr[j])) {
                j++;
            }
            tokens.push_back({TOKEN_NUMBER, stoi(expr.substr(i, j - i))});
            i = j;
        } else if (expr[i] == '+') {
            tokens.push_back({TOKEN_PLUS, 0});
            i++;
        } else if (expr[i] == '-') {
            tokens.push_back({TOKEN_MINUS, 0});
            i++;
        } else if (expr[i] == '*') {
            tokens.push_back({TOKEN_MULTIPLY, 0});
            i++;
        } else if (expr[i] == '/') {
            tokens.push_back({TOKEN_DIVIDE, 0});
            i++;
        } else if (expr[i] == '(') {
            tokens.push_back({TOKEN_LPAREN, 0});
            i++;
        } else if (expr[i] == ')') {
            tokens.push_back({TOKEN_RPAREN, 0});
            i++;
        } else {
            i++;
        }
    }
    return tokens;
}

// 使用算法分析表达式
bool analyzeExpression(string expr) {
    vector<Token> tokens = expressionToTokens(expr);
    tokens.push_back({TOKEN_NUMBER, 0});
    stack<int> state_stack;
    stack<TokenType> symbol_stack;
    state_stack.push(0);
    symbol_stack.push(TOKEN_NUMBER);
    size_t i = 0;
    while (!state_stack.empty()) {
        int state = state_stack.top();
        TokenType symbol = symbol_stack.top();
        if (action.count({state, symbol}) == 0) {
            return false;
        }
        char action_type = action[{state, symbol}].first;
        int action_param = action[{state, symbol}].second;
        if (action_type == 's') {
            state_stack.push(action_param);
            symbol_stack.push(symbol);
            state_stack.push(0);
            symbol_stack.push(tokens[i].type);
            i++;
        } else if (action_type == 'r') {
            Production production = productions[action_param];
            for (size_t j = 0; j < production.right.size(); j++) {
                state_stack.pop();
                symbol_stack.pop();
            }
            int top_state = state_stack.top();
            TokenType top_symbol = symbol_stack.top();
            if (goto_table.count({top_state, production.left}) == 0) {
                return false;
            }
            int next_state = goto_table[{top_state, production.left}];
            state_stack.push(next_state);
            symbol_stack.push(production.left);
        } else if (action_type == 'a') {
            return true;
        } else {
            return false;
        }
    }
    return false;
}

int main() {
    calculateItemSets();
    calculateTables();
    cout << actionTableToString() << endl;
    cout << gotoTableToString() << endl;
    string expr;
    cout << "Enter an expression: ";
    cin >> expr;
    if (analyzeExpression(expr)) {
        vector<Token> tokens = expressionToTokens(expr);
        cout << "Result: " << calculate(tokens) << endl;
    } else {
        cout << "ERROR: Invalid expression!\n";
    }
    return 0;
}

这里使用了一个简单的算数表达式语法，只包含加、减、乘、除和括号运算，并且没有考虑优先级和结合性。算法输出 ACTION 表和 GOTO 表，并且可以解析输入的表达式并计算其值。