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

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:

Construct FA for {w | w $\epsilon$ {a, b}} and Na = 0 mod 3 means having a’s as a multiple of 3.
Construct FA for {w | w $\epsilon$ {a, b}} and Nb = 0 mod 3 means having b’s as a multiple of 3.
Concatenate the two FA and make single DFA using Concatenation process in DFA.

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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