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

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 {a, b}} and Na = 0 mod 3 means having a’s as a multiple of 3.
2. Construct FA for {w | w {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.

1. 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  #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

Output:

ACCEPTED

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Improved By : Rajput-Ji, 29AjayKumar