构造一个接受语言 L = {w | 的 DFA w ∈ {a,b}* 和 Na(w) mod 3 = Nb (w) mod 3}

问题：构造一个确定性有限自动机 (DFA) 来接受语言 L = {w |赢; {a,b}* 和 Na(w) mod 3 = Nb (w) mod 3}。

语言 L={w | Na(w) = Nb(w)mod 3} 这意味着所有字符串都包含 a 的计数模数等于 b 的计数模数乘以 3。

例子：

Input: a a b b b
Output: NOT ACCEPTED
// n = 2, m=3 (2 mod 3! = 3 mod 3)

Input: a a b b
Output: ACCEPTED
// n = 2, m = 2 (2 mod 3 = 2 mod 3)

Input: b b b
Output: NOT ACCEPTED
// n = 0, m = 3 ((0 mod 3 = 3 mod 3) <=> (0=0))

编程需要懂一点英语

方法：

为 {w | 构造 FA瓦 $\epsilon$ {a, b}} 和 Na = 0 mod 3 表示 a 是 3 的倍数。
为 {w | 构造 FA瓦 $\epsilon$ {a, b}} 和 Nb = 0 mod 3 表示 b 是 3 的倍数。
连接两个 FA 并使用 DFA 中的连接过程制作单个 DFA。

将状态作为接受 a 和 b 的相等模数计数的最终状态。

DFA 状态转换图：

状态 00、11 和 22 导致接受字符串。而状态 01、02、10、12、20 和 21 导致拒绝字符串。

让我们看看演示的代码：

C/C++

// C/C++ Program to construct a DFA which accept the language
// L = { w | w is in {a, b}* and Na(w) mod 3 = Nb(w) mod 3}
#include 
#include 
  
// dfa tells the number associated
// string end in which state.
int dfa = 0;
  
// This function is for
// the starting state (00)of DFA
void start(char c)
{
    if (c == 'a') {
        dfa = 10;
    }
    else if (c == 'b') {
        dfa = 1;
    }
  
    // -1 is used to check for any invalid symbol
    else {
        dfa = -1;
    }
}
  
// This function is for the first state (01) of DFA
void state01(char c)
{
    if (c == 'a') {
        dfa = 11;
    }
    else if (c == 'b') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}
  
// This function is for the second state (02) of DFA
void state02(char c)
{
    if (c == 'b') {
        dfa = 0;
    }
    else if (c == 'a') {
        dfa = 12;
    }
    else {
        dfa = -1;
    }
}
  
// This function is for the third state (10)of DFA
void state10(char c)
{
    if (c == 'b') {
        dfa = 11;
    }
    else if (c == 'a') {
        dfa = 20;
    }
    else {
        dfa = -1;
    }
}
  
// This function is for the forth state (11)of DFA
void state11(char c)
{
    if (c == 'b') {
        dfa = 12;
    }
    else if (c == 'a') {
        dfa = 21;
    }
    else {
        dfa = -1;
    }
}
  
void state12(char c)
{
    if (c == 'b') {
        dfa = 10;
    }
    else if (c == 'a') {
        dfa = 22;
    }
    else {
        dfa = -1;
    }
}
  
void state20(char c)
{
    if (c == 'b') {
        dfa = 21;
    }
    else if (c == 'a') {
        dfa = 0;
    }
    else {
        dfa = -1;
    }
}
  
void state21(char c)
{
    if (c == 'b') {
        dfa = 22;
    }
    else if (c == 'a') {
        dfa = 1;
    }
    else {
        dfa = -1;
    }
}
  
void state22(char c)
{
    if (c == 'b') {
        dfa = 20;
    }
    else if (c == 'a') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}
  
int isAccepted(char str[])
{
    // store length of string
    int i, len = strlen(str);
  
    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);
  
        else if (dfa == 1)
            state01(str[i]);
  
        else if (dfa == 2)
            state02(str[i]);
  
        else if (dfa == 10)
            state10(str[i]);
  
        else if (dfa == 11)
            state11(str[i]);
  
        else if (dfa == 12)
            state12(str[i]);
  
        else if (dfa == 20)
            state20(str[i]);
  
        else if (dfa == 21)
            state21(str[i]);
  
        else if (dfa == 22)
            state22(str[i]);
  
        else
            return 0;
    }
    if (dfa == 0 || dfa == 11 || dfa == 22)
        return 1;
    else
        return 0;
}
  
// driver code
int main()
{
    char str[] = "aaaabbbb";
    if (isAccepted(str))
        printf("ACCEPTED");
    else
        printf("NOT ACCEPTED");
    return 0;
}
  
// This code is contributed by SHUBHAMSINGH10.

Java

// Java  Program to construct a DFA which accept the language
// L = { w | w is in {a, b}* and Na(w) mod 3 = Nb(w) mod 3}
class GFG{
// dfa tells the number associated
// string end in which state.
static int dfa = 0;
   
// This function is for
// the starting state (00)of DFA
static void start(char c)
{
    if (c == 'a') {
        dfa = 10;
    }
    else if (c == 'b') {
        dfa = 1;
    }
   
    // -1 is used to check for any invalid symbol
    else {
        dfa = -1;
    }
}
   
// This function is for the first state (01) of DFA
static void state01(char c)
{
    if (c == 'a') {
        dfa = 11;
    }
    else if (c == 'b') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}
   
// This function is for the second state (02) of DFA
static void state02(char c)
{
    if (c == 'b') {
        dfa = 0;
    }
    else if (c == 'a') {
        dfa = 12;
    }
    else {
        dfa = -1;
    }
}
   
// This function is for the third state (10)of DFA
static void state10(char c)
{
    if (c == 'b') {
        dfa = 11;
    }
    else if (c == 'a') {
        dfa = 20;
    }
    else {
        dfa = -1;
    }
}
   
// This function is for the forth state (11)of DFA
static void state11(char c)
{
    if (c == 'b') {
        dfa = 12;
    }
    else if (c == 'a') {
        dfa = 21;
    }
    else {
        dfa = -1;
    }
}
   
static void state12(char c)
{
    if (c == 'b') {
        dfa = 10;
    }
    else if (c == 'a') {
        dfa = 22;
    }
    else {
        dfa = -1;
    }
}
   
static void state20(char c)
{
    if (c == 'b') {
        dfa = 21;
    }
    else if (c == 'a') {
        dfa = 0;
    }
    else {
        dfa = -1;
    }
}
   
static void state21(char c)
{
    if (c == 'b') {
        dfa = 22;
    }
    else if (c == 'a') {
        dfa = 1;
    }
    else {
        dfa = -1;
    }
}
   
static void state22(char c)
{
    if (c == 'b') {
        dfa = 20;
    }
    else if (c == 'a') {
        dfa = 2;
    }
    else {
        dfa = -1;
    }
}
   
static boolean isAccepted(String st)
{
    // store length of string
    int i, len = st.length();
    char[] str = st.toCharArray();
    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);
   
        else if (dfa == 1)
            state01(str[i]);
   
        else if (dfa == 2)
            state02(str[i]);
   
        else if (dfa == 10)
            state10(str[i]);
   
        else if (dfa == 11)
            state11(str[i]);
   
        else if (dfa == 12)
            state12(str[i]);
   
        else if (dfa == 20)
            state20(str[i]);
   
        else if (dfa == 21)
            state21(str[i]);
   
        else if (dfa == 22)
            state22(str[i]);
   
        else
            return false;
    }
    if (dfa == 0 || dfa == 11 || dfa == 22)
        return true;
    else
        return false;
}
   
// driver code
public static void main(String []args)
    {
    String str = "aaaabbbb";
    if (isAccepted(str))
        System.out.println("ACCEPTED");
    else
        System.out.println("NOT ACCEPTED");
    }
}
  
// This code contributed by Rajput-Ji

Python 3

"""Python Program to construct a DFA which accept the language 
L =  {w | w belongs to {a, b}*}
and Na(w)mod3 = Nb(w)mod 3"""
    
# This function is for 
# the starting state (00)of DFA 
def start( c): 
   
    if (c == 'a'):  
        dfa = 10 
       
    elif (c == 'b') : 
        dfa = 1 
       
    
     #-1 is used to check for any invalid symbol 
    else:  
        dfa = -1
    return dfa
       
   
    
# This function is for the first state (01) of DFA 
def state01( c): 
   
    if (c == 'a'):  
        dfa = 11 
       
    elif (c == 'b') : 
        dfa = 2 
       
    else:  
        dfa = -1
    return dfa
   
    
# This function is for the second state (02) of DFA 
def state02( c): 
   
    if (c == 'b') : 
        dfa = 0 
       
    elif(c =='a'):
      
        dfa = 12
      
    else:  
        dfa = -1
    return dfa
       
   
    
# This function is for the third state (10)of DFA 
def state10( c): 
   
    if (c == 'b') : 
        dfa = 11 
       
    elif(c =='a'):
      
        dfa = 20
      
    else:  
        dfa = -1
    return dfa
       
   
  
# This function is for the forth state (11)of DFA 
def state11( c): 
   
    if (c == 'b') : 
        dfa = 12 
       
    elif(c =='a'):
      
        dfa = 21
      
    else:  
        dfa = -1
    return dfa
       
   
  
def state12( c): 
   
    if (c == 'b') : 
        dfa = 10 
       
    elif(c =='a'):
      
        dfa = 22
      
    else:  
        dfa = -1
    return dfa
       
   
  
def state20( c): 
   
    if (c == 'b') : 
        dfa = 21 
       
    elif(c =='a'):
      
        dfa = 0
      
    else:  
        dfa = -1
    return dfa
       
  
  
def state21( c): 
   
    if (c == 'b') : 
        dfa = 22 
       
    elif(c =='a'):
      
        dfa = 1
      
    else:  
        dfa = -1
    return dfa
  
  
def state22( c): 
   
    if (c == 'b') : 
        dfa = 20 
       
    elif(c =='a'):
      
        dfa = 2
      
    else:  
        dfa = -1
    return dfa
       
  
  
def isAccepted(str): 
   
    # store length of string 
    l = len(str)
    # dfa tells the number associated  
    # with the present dfa = state  
    dfa = 0
      
    for i in range(l):  
        if (dfa == 0) :
            start(str[i]) 
    
        elif (dfa == 1): 
            state01(str[i]) 
    
        elif (dfa == 2) :
            state02(str[i]) 
    
        elif (dfa == 10) :
            state10(str[i]) 
    
        elif (dfa == 11) :
            state11(str[i]) 
              
        elif (dfa == 12) :
            state12(str[i]) 
          
        elif (dfa == 20) :
            state20(str[i]) 
    
        elif (dfa == 21) :
            state21(str[i]) 
    
        elif (dfa == 22) :
            state22(str[i]) 
              
        else:
            return 0 
       
    if (dfa == 0 or dfa == 11 or dfa == 22) :
        return 1 
    else:
        return 0 
   
    
# Driver code   
if __name__ == "__main__" : 
   
    string = "aaaabbbb" 
    if (isAccepted(string)) :
        print("ACCEPTED") 
    else:
        print("NOT ACCEPTED") 
   
  
# This code is contributed by SHUBHAMSINGH10.

C#

// C# Program to construct a DFA which accept the language 
// L = { w | w is in {a, b}* and Na(w) mod 3 = Nb(w) mod 3} 
using System;
  
class GFG
{ 
  
// DFA tells the number associated 
// string end in which state. 
static int dfa = 0; 
  
// This function is for 
// the starting state (00)of DFA 
static void start(char c) 
{ 
    if (c == 'a') { 
        dfa = 10; 
    } 
    else if (c == 'b') { 
        dfa = 1; 
    } 
  
    // -1 is used to check for any invalid symbol 
    else { 
        dfa = -1; 
    } 
} 
  
// This function is for the first state (01) of DFA 
static void state01(char c) 
{ 
    if (c == 'a') { 
        dfa = 11; 
    } 
    else if (c == 'b') { 
        dfa = 2; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
// This function is for the second state (02) of DFA 
static void state02(char c) 
{ 
    if (c == 'b') { 
        dfa = 0; 
    } 
    else if (c == 'a') { 
        dfa = 12; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
// This function is for the third state (10)of DFA 
static void state10(char c) 
{ 
    if (c == 'b') { 
        dfa = 11; 
    } 
    else if (c == 'a') { 
        dfa = 20; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
// This function is for the forth state (11)of DFA 
static void state11(char c) 
{ 
    if (c == 'b') { 
        dfa = 12; 
    } 
    else if (c == 'a') { 
        dfa = 21; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
static void state12(char c) 
{ 
    if (c == 'b') { 
        dfa = 10; 
    } 
    else if (c == 'a') { 
        dfa = 22; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
static void state20(char c) 
{ 
    if (c == 'b') { 
        dfa = 21; 
    } 
    else if (c == 'a') { 
        dfa = 0; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
static void state21(char c) 
{ 
    if (c == 'b') { 
        dfa = 22; 
    } 
    else if (c == 'a') { 
        dfa = 1; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
static void state22(char c) 
{ 
    if (c == 'b') { 
        dfa = 20; 
    } 
    else if (c == 'a') { 
        dfa = 2; 
    } 
    else { 
        dfa = -1; 
    } 
} 
  
static Boolean isAccepted(String st) 
{ 
    // store length of string 
    int i, len = st.Length; 
    char[] str = st.ToCharArray(); 
    for (i = 0; i < len; i++) { 
        if (dfa == 0) 
            start(str[i]); 
  
        else if (dfa == 1) 
            state01(str[i]); 
  
        else if (dfa == 2) 
            state02(str[i]); 
  
        else if (dfa == 10) 
            state10(str[i]); 
  
        else if (dfa == 11) 
            state11(str[i]); 
  
        else if (dfa == 12) 
            state12(str[i]); 
  
        else if (dfa == 20) 
            state20(str[i]); 
  
        else if (dfa == 21) 
            state21(str[i]); 
  
        else if (dfa == 22) 
            state22(str[i]); 
  
        else
            return false; 
    } 
    if (dfa == 0 || dfa == 11 || dfa == 22) 
        return true; 
    else
        return false; 
} 
  
// Driver code 
public static void Main(String []args) 
{ 
    String str = "aaaabbbb"; 
    if (isAccepted(str)) 
        Console.WriteLine("ACCEPTED"); 
    else
        Console.WriteLine("NOT ACCEPTED"); 
} 
} 
  
// This code is contributed by 29AjayKumar

输出：

ACCEPTED