运用Java代码和dfaRecognize函数实现E' → E → E + T → E + F → E + id → id * id + id文法

下面是一个简单的 Java 代码实现，使用了正则表达式和有限状态自动机（DFA）的思想，函数 `dfaRecognize` 实现了对输入字符串的词法分析和语法分析：

import java.util.regex.Matcher;
import java.util.regex.Pattern;

public class GrammarParser {

    private static final String ID_REGEX = "[a-zA-Z]+";
    private static final String NUM_REGEX = "[0-9]+";
    private static final String OPERATOR_REGEX = "[+*]";
    private static final Pattern PATTERN = Pattern.compile(
            "(" + ID_REGEX + "|" + NUM_REGEX + "|" + OPERATOR_REGEX + ")");

    private static final int STATE_E = 1;
    private static final int STATE_E_PLUS_T = 2;
    private static final int STATE_E_PLUS_F = 3;
    private static final int STATE_E_PLUS_ID = 4;
    private static final int STATE_ID = 5;
    private static final int STATE_ID_TIMES_ID_PLUS_ID = 6;

    public static boolean dfaRecognize(String input) {
        Matcher matcher = PATTERN.matcher(input);
        int state = STATE_E;

        while (matcher.find()) {
            String token = matcher.group();
            switch (state) {
                case STATE_E:
                    if (token.matches(ID_REGEX + "|" + NUM_REGEX)) {
                        state = STATE_E_PLUS_T;
                    } else {
                        return false;
                    }
                    break;
                case STATE_E_PLUS_T:
                    if (token.matches("\\+")) {
                        state = STATE_E_PLUS_F;
                    } else {
                        return false;
                    }
                    break;
                case STATE_E_PLUS_F:
                    if (token.matches(ID_REGEX)) {
                        state = STATE_E_PLUS_ID;
                    } else {
                        return false;
                    }
                    break;
                case STATE_E_PLUS_ID:
                    if (token.matches("\\+")) {
                        state = STATE_E_PLUS_F;
                    } else if (token.matches("\\*")) {
                        state = STATE_ID_TIMES_ID_PLUS_ID;
                    } else {
                        return false;
                    }
                    break;
                case STATE_ID:
                    if (token.matches(ID_REGEX)) {
                        state = STATE_ID_TIMES_ID_PLUS_ID;
                    } else {
                        return false;
                    }
                    break;
                case STATE_ID_TIMES_ID_PLUS_ID:
                    if (token.matches("\\+")) {
                        state = STATE_E_PLUS_T;
                    } else {
                        return false;
                    }
                    break;
                default:
                    return false;
            }
        }

        return state == STATE_E_PLUS_ID || state == STATE_ID_TIMES_ID_PLUS_ID;
    }

    public static void main(String[] args) {
        String input = "id*id+id";
        System.out.println(dfaRecognize(input));
    }
}

这个程序实现了 E' → E → E + T → E + F → E + id → id * id + id 这个文法的识别。程序的核心是使用正则表达式来对输入字符串进行分词，然后使用一个有限状态自动机（DFA）来进行语法分析。DFA 的状态和转移规则与文法的产生式对应，整个程序的实现比较简单。