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Construct a DFA which accept the language L = {w | w ∈ {a,b}* and Na(w) mod 3 = Nb (w) mod 3}

  • Difficulty Level : Hard
  • Last Updated : 02 Jul, 2019
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Problem: Construct a deterministic finite automata (DFA) for accepting the language L = {w | w ∈ {a,b}* and Na(w) mod 3 = Nb (w) mod 3}.

Language L={w | Na(w) = Nb(w)mod 3} which means all string contain modulus of count of a’s equal to modulus of count of b’s by 3.

Examples:

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))

Approaches:

  1. Construct FA for {w | w \epsilon {a, b}} and Na = 0 mod 3 means having a’s as a multiple of 3.
  2. Construct FA for {w | w \epsilon {a, b}} and Nb = 0 mod 3 means having b’s as a multiple of 3.
  3. Concatenate the two FA and make single DFA using Concatenation process in DFA.
  4. Making state as final state which accepts the equal modulus count for a’s and b’s.

      DFA State Transition Diagram:

      States 00, 11 and 22 leads to the acceptance of the string.Whereas states 01, 02, 10, 12, 20 and 21 leads to the rejection of the string.

      Let’s see code for the demonstration:

      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 <stdio.h>
      #include <string.h>
        
      // 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

      Output:

      ACCEPTED

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