NFA to accept strings that has atleast one character occurring in a multiple of 3

Prerequisites: Finite Automata

Given a string str consisting of characters a, b and c, check if the number of occurances of any character in the string is a multiple of 3 or not.

Examples:



Input: str = bc
Output: ACCEPTED
Explanation: The string consists 0 a’s and 3 * 0 = 0.

Input: str = abccc
Output: ACCEPTED
Explanation: The string consists 3 c’s.

Input: str = abc
Output: NOT ACCEPTED

Approach:
An NFA or a Nondeterministic Finite Automata is very similar to a DFA. It is a finite state machine which accepts a string(under some specific condition) if it reaches a final state, otherwise rejects it. The additional features which an NFA has are:

  1. Null move is allowed i.e., it can move forward without reading symbols.
  2. Ability to transmit to any number of states for a particular input.

NFA Machine that accepts all strings in which the occurrences of atleast one character is a multiple of 3:
For the above problem statement, we must first build an NFA machine. NFA machine is similar to a flowchart with various states and transitions. NFA machine corresponding to the above problem is shown below, Q3, Q4 and Q8 are the final states:

How does this NFA Machine work:
The working of the machine depends on checking if the string has 3 multiples of a’s or b’s or c’s.

  • Case 1: Number of a’s is a multiple of three:
    • To check whether the number of a’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q2, Q3, Q4 check whether the number of a’s is a multiple of three or not. If at any point this case reaches the final state Q2, then the number of a’s is a multiple of three.
  • Case 2: Number of b’s is a multiple of three:
    • To check whether the number of b’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q5, Q6, Q7 check whether the number of b’s is a multiple of three or not. If at any point this case reaches the final state Q5, then the number of b’s is a multiple of three.
  • Case 3: Number of c’s is a multiple of three:
    • To check whether the number of c’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q8, Q9, Q10 check whether the number of c’s is a multiple of three or not. If at any point this case reaches the final state Q8, then the number of c’s is a multiple of three.
  • If any of the above mentioned states reach the final states Q2, Q5 or Q8; then the string satisfies the condition

Below is the implementation of the above approach:

C++

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// C++ implementation of the above approach
#include <bits/stdc++.h>
  
// NFA variable that keeps track of
// the state while transaction.
int nfa = 1;
  
// This checks for invalid input.
int flag = 0;
using namespace std;
  
// Function for the state Q2
void state1(char c)
{
    // State transitions
    // 'a' takes to Q4, and
    // 'b' and 'c' remain at Q2
    if (c == 'a')
        nfa = 2;
    else if (c == 'b' || c == 'c')
        nfa = 1;
    else
        flag = 1;
}
  
// Function for the state Q3
void state2(char c)
{
    // State transitions
    // 'a' takes to Q3, and
    // 'b' and 'c' remain at Q4
    if (c == 'a')
        nfa = 3;
    else if (c == 'b' || c == 'c')
        nfa = 2;
    else
        flag = 1;
}
  
// Function for the state Q4
void state3(char c)
{
    // State transitions
    // 'a' takes to Q2, and
    // 'b' and 'c' remain at Q3
    if (c == 'a')
        nfa = 1;
    else if (c == 'b' || c == 'c')
        nfa = 3;
    else
        flag = 1;
}
  
// Function for the state Q5
void state4(char c)
{
    // State transitions
    // 'b' takes to Q6, and
    // 'a' and 'c' remain at Q5
    if (c == 'b')
        nfa = 5;
    else if (c == 'a' || c == 'c')
        nfa = 4;
    else
        flag = 1;
}
  
// Function for the state Q6
void state5(char c)
{
    // State transitions
    // 'b' takes to Q7, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 6;
    else if (c == 'a' || c == 'c')
        nfa = 5;
    else
        flag = 1;
}
  
// Function for the state Q7
void state6(char c)
{
    // State transitions
    // 'b' takes to Q5, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 4;
    else if (c == 'a' || c == 'c')
        nfa = 6;
    else
        flag = 1;
}
  
// Function for the state Q8
void state7(char c)
{
    // State transitions
    // 'c' takes to Q9, and
    // 'a' and 'b' remain at Q8
    if (c == 'c')
        nfa = 8;
    else if (c == 'b' || c == 'a')
        nfa = 7;
    else
        flag = 1;
}
  
// Function for the state Q9
void state8(char c)
{
    // State transitions
    // 'c' takes to Q10, and
    // 'a' and 'b' remain at Q9
    if (c == 'c')
        nfa = 9;
    else if (c == 'b' || c == 'a')
        nfa = 8;
    else
        flag = 1;
}
  
// Function for the state Q10
void state9(char c)
{
    // State transitions
    // 'c' takes to Q8, and
    // 'a' and 'b' remain at Q10
    if (c == 'c')
        nfa = 7;
    else if (c == 'b' || c == 'a')
        nfa = 9;
    else
        flag = 1;
}
  
// Function to check for 3 a's
bool checkA(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 1)
            state1(s[i]);
        else if (nfa == 2)
            state2(s[i]);
        else if (nfa == 3)
            state3(s[i]);
    }
    if (nfa == 1) {
        return true;
    }
    else {
        nfa = 4;
    }
}
  
// Function to check for 3 b's
bool checkB(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 4)
            state4(s[i]);
        else if (nfa == 5)
            state5(s[i]);
        else if (nfa == 6)
            state6(s[i]);
    }
    if (nfa == 4) {
  
        return true;
    }
    else {
        nfa = 7;
    }
}
  
// Function to check for 3 c's
bool checkC(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 7)
            state7(s[i]);
        else if (nfa == 8)
            state8(s[i]);
        else if (nfa == 9)
            state9(s[i]);
    }
    if (nfa == 7) {
  
        return true;
    }
}
  
// Driver Code
int main()
{
    string s = "bbbca";
    int x = 5;
  
    // If any of the states is true, that is, if either
    // the number of a's or number of b's or number of c's
    // is a multiple of three, then the string is accepted
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) {
        cout << "ACCEPTED";
    }
  
    else {
        if (flag == 0) {
            cout << "NOT ACCEPTED";
            return 0;
        }
        else {
            cout << "INPUT OUT OF DICTIONARY.";
            return 0;
        }
    }
}

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// Java implementation of the above approach 
class GFG {
      
// NFA variable that keeps track of 
// the state while transaction. 
static int nfa = 1
  
// This checks for invalid input. 
static int flag = 0
  
// Function for the state Q2 
static void state1(char c) 
    // State transitions 
    // 'a' takes to Q4, and 
    // 'b' and 'c' remain at Q2 
    if (c == 'a'
        nfa = 2
    else if (c == 'b' || c == 'c'
        nfa = 1
    else
        flag = 1
  
// Function for the state Q3 
static void state2(char c) 
    // State transitions 
    // 'a' takes to Q3, and 
    // 'b' and 'c' remain at Q4 
    if (c == 'a'
        nfa = 3
    else if (c == 'b' || c == 'c'
        nfa = 2
    else
        flag = 1
  
// Function for the state Q4 
static void state3(char c) 
    // State transitions 
    // 'a' takes to Q2, and 
    // 'b' and 'c' remain at Q3 
    if (c == 'a'
        nfa = 1
    else if (c == 'b' || c == 'c'
        nfa = 3
    else
        flag = 1
  
// Function for the state Q5 
static void state4(char c) 
    // State transitions 
    // 'b' takes to Q6, and 
    // 'a' and 'c' remain at Q5 
    if (c == 'b'
        nfa = 5
    else if (c == 'a' || c == 'c'
        nfa = 4
    else
        flag = 1
  
// Function for the state Q6 
static void state5(char c) 
    // State transitions 
    // 'b' takes to Q7, and 
    // 'a' and 'c' remain at Q7 
    if (c == 'b'
        nfa = 6
    else if (c == 'a' || c == 'c'
        nfa = 5
    else
        flag = 1
  
// Function for the state Q7 
static void state6(char c) 
    // State transitions 
    // 'b' takes to Q5, and 
    // 'a' and 'c' remain at Q7 
    if (c == 'b'
        nfa = 4
    else if (c == 'a' || c == 'c'
        nfa = 6
    else
        flag = 1
  
// Function for the state Q8 
static void state7(char c) 
    // State transitions 
    // 'c' takes to Q9, and 
    // 'a' and 'b' remain at Q8 
    if (c == 'c'
        nfa = 8
    else if (c == 'b' || c == 'a'
        nfa = 7
    else
        flag = 1
  
// Function for the state Q9 
static void state8(char c) 
    // State transitions 
    // 'c' takes to Q10, and 
    // 'a' and 'b' remain at Q9 
    if (c == 'c'
        nfa = 9
    else if (c == 'b' || c == 'a'
        nfa = 8
    else
        flag = 1
  
// Function for the state Q10 
static void state9(char c) 
    // State transitions 
    // 'c' takes to Q8, and 
    // 'a' and 'b' remain at Q10 
    if (c == 'c'
        nfa = 7
    else if (c == 'b' || c == 'a'
        nfa = 9
    else
        flag = 1
  
// Function to check for 3 a's 
static boolean checkA(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 1
            state1(s.charAt(i)); 
        else if (nfa == 2
            state2(s.charAt(i)); 
        else if (nfa == 3
            state3(s.charAt(i)); 
    
    if (nfa == 1) { 
        return true
    
    else
        nfa = 4
    
    return false;
  
// Function to check for 3 b's 
static boolean checkB(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 4
            state4(s.charAt(i)); 
        else if (nfa == 5
            state5(s.charAt(i)); 
        else if (nfa == 6
            state6(s.charAt(i)); 
    
    if (nfa == 4) { 
  
        return true
    
    else
        nfa = 7
    
    return false;
  
// Function to check for 3 c's 
static boolean checkC(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 7
            state7(s.charAt(i)); 
        else if (nfa == 8
            state8(s.charAt(i)); 
        else if (nfa == 9
            state9(s.charAt(i)); 
    
    if (nfa == 7) { 
  
        return true
    
    return false;
  
// Driver Code 
public static void main (String[] args)
    String s = "bbbca"
    int x = 5
  
    // If any of the states is true, that is, if either 
    // the number of a's or number of b's or number of c's 
    // is a multiple of three, then the string is accepted 
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) { 
        System.out.println("ACCEPTED"); 
    
  
    else
        if (flag == 0) { 
            System.out.println("NOT ACCEPTED"); 
              
        
        else
            System.out.println("INPUT OUT OF DICTIONARY."); 
              
        
    
}
  
// This code is contributed by AnkitRai01

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Python3

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# Python3 implementation of the above approach
  
# NFA variable that keeps track of
# the state while transaction.
nfa = 1
  
# This checks for invalid input.
flag = 0
  
# Function for the state Q2
def state1(c):
    global nfa,flag
  
    # State transitions
    # 'a' takes to Q4, and
    # 'b' and 'c' remain at Q2
    if (c == 'a'):
        nfa = 2
    elif (c == 'b' or c == 'c'):
        nfa = 1
    else:
        flag = 1
  
# Function for the state Q3
def state2(c):
    global nfa,flag
  
    # State transitions
    # 'a' takes to Q3, and
    # 'b' and 'c' remain at Q4
    if (c == 'a'):
        nfa = 3
    elif (c == 'b' or c == 'c'):
        nfa = 2
    else:
        flag = 1
  
# Function for the state Q4
def state3(c):
    global nfa,flag
  
    # State transitions
    # 'a' takes to Q2, and
    # 'b' and 'c' remain at Q3
    if (c == 'a'):
        nfa = 1
    elif (c == 'b' or c == 'c'):
        nfa = 3
    else:
        flag = 1
  
# Function for the state Q5
def state4(c):
    global nfa,flag
  
    # State transitions
    # 'b' takes to Q6, and
    # 'a' and 'c' remain at Q5
    if (c == 'b'):
        nfa = 5
    elif (c == 'a' or c == 'c'):
        nfa = 4
    else:
        flag = 1
  
# Function for the state Q6
def state5(c):
    global nfa, flag
  
    # State transitions
    # 'b' takes to Q7, and
    # 'a' and 'c' remain at Q7
    if (c == 'b'):
        nfa = 6
    elif (c == 'a' or c == 'c'):
        nfa = 5
    else:
        flag = 1
  
# Function for the state Q7
def state6(c):
    global nfa,flag
  
    # State transitions
    # 'b' takes to Q5, and
    # 'a' and 'c' remain at Q7
    if (c == 'b'):
        nfa = 4
    elif (c == 'a' or c == 'c'):
        nfa = 6
    else:
        flag = 1
  
# Function for the state Q8
def state7(c):
    global nfa,flag
  
    # State transitions
    # 'c' takes to Q9, and
    # 'a' and 'b' remain at Q8
    if (c == 'c'):
        nfa = 8
    elif (c == 'b' or c == 'a'):
        nfa = 7
    else:
        flag = 1
  
# Function for the state Q9
def state8(c):
    global nfa,flag
  
    # State transitions
    # 'c' takes to Q10, and
    # 'a' and 'b' remain at Q9
    if (c == 'c'):
        nfa = 9
    elif (c == 'b' or c == 'a'):
        nfa = 8
    else:
        flag = 1
  
# Function for the state Q10
def state9(c):
    global nfa,flag
  
    # State transitions
    # 'c' takes to Q8, and
    # 'a' and 'b' remain at Q10
    if (c == 'c'):
        nfa = 7
    elif (c == 'b' or c == 'a'):
        nfa = 9
    else:
        flag = 1
          
# Function to check for 3 a's
def checkA(s, x):
    global nfa,flag
    for i in range(x):
        if (nfa == 1):
            state1(s[i])
        elif (nfa == 2):
            state2(s[i])
        elif (nfa == 3):
            state3(s[i])
      
    if (nfa == 1): 
        return True
      
    else
        nfa = 4
      
# Function to check for 3 b's
def checkB(s, x):
    global nfa,flag
    for i in range(x): 
        if (nfa == 4):
            state4(s[i])
        elif (nfa == 5):
            state5(s[i])
        elif (nfa == 6):
            state6(s[i])
      
    if (nfa == 4):
        return True
    else
        nfa = 7
      
# Function to check for 3 c's
def checkC(s, x):
    global nfa, flag
    for i in range(x):
        if (nfa == 7):
            state7(s[i])
        elif (nfa == 8):
            state8(s[i])
        elif (nfa == 9):
            state9(s[i])
              
    if (nfa == 7):
        return True
  
# Driver Code
  
s = "bbbca"
x = 5
  
# If any of the states is True, that is, if either
# the number of a's or number of b's or number of c's
# is a multiple of three, then the is accepted
if (checkA(s, x) or checkB(s, x) or checkC(s, x)):
    print("ACCEPTED")
  
else:
    if (flag == 0):
        print("NOT ACCEPTED")
      
    else:
        print("INPUT OUT OF DICTIONARY.")
          
# This code is contributed by shubhamsingh10

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C#

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// C# implementation of the above approach 
using System;
  
class GFG {
       
// NFA variable that keeps track of 
// the state while transaction. 
static int nfa = 1; 
   
// This checks for invalid input. 
static int flag = 0; 
   
// Function for the state Q2 
static void state1(char c) 
    // State transitions 
    // 'a' takes to Q4, and 
    // 'b' and 'c' remain at Q2 
    if (c == 'a'
        nfa = 2; 
    else if (c == 'b' || c == 'c'
        nfa = 1; 
    else
        flag = 1; 
   
// Function for the state Q3 
static void state2(char c) 
    // State transitions 
    // 'a' takes to Q3, and 
    // 'b' and 'c' remain at Q4 
    if (c == 'a'
        nfa = 3; 
    else if (c == 'b' || c == 'c'
        nfa = 2; 
    else
        flag = 1; 
   
// Function for the state Q4 
static void state3(char c) 
    // State transitions 
    // 'a' takes to Q2, and 
    // 'b' and 'c' remain at Q3 
    if (c == 'a'
        nfa = 1; 
    else if (c == 'b' || c == 'c'
        nfa = 3; 
    else
        flag = 1; 
   
// Function for the state Q5 
static void state4(char c) 
    // State transitions 
    // 'b' takes to Q6, and 
    // 'a' and 'c' remain at Q5 
    if (c == 'b'
        nfa = 5; 
    else if (c == 'a' || c == 'c'
        nfa = 4; 
    else
        flag = 1; 
   
// Function for the state Q6 
static void state5(char c) 
    // State transitions 
    // 'b' takes to Q7, and 
    // 'a' and 'c' remain at Q7 
    if (c == 'b'
        nfa = 6; 
    else if (c == 'a' || c == 'c'
        nfa = 5; 
    else
        flag = 1; 
   
// Function for the state Q7 
static void state6(char c) 
    // State transitions 
    // 'b' takes to Q5, and 
    // 'a' and 'c' remain at Q7 
    if (c == 'b'
        nfa = 4; 
    else if (c == 'a' || c == 'c'
        nfa = 6; 
    else
        flag = 1; 
   
// Function for the state Q8 
static void state7(char c) 
    // State transitions 
    // 'c' takes to Q9, and 
    // 'a' and 'b' remain at Q8 
    if (c == 'c'
        nfa = 8; 
    else if (c == 'b' || c == 'a'
        nfa = 7; 
    else
        flag = 1; 
   
// Function for the state Q9 
static void state8(char c) 
    // State transitions 
    // 'c' takes to Q10, and 
    // 'a' and 'b' remain at Q9 
    if (c == 'c'
        nfa = 9; 
    else if (c == 'b' || c == 'a'
        nfa = 8; 
    else
        flag = 1; 
   
// Function for the state Q10 
static void state9(char c) 
    // State transitions 
    // 'c' takes to Q8, and 
    // 'a' and 'b' remain at Q10 
    if (c == 'c'
        nfa = 7; 
    else if (c == 'b' || c == 'a'
        nfa = 9; 
    else
        flag = 1; 
   
// Function to check for 3 a's 
static bool checkA(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 1) 
            state1(s[i]); 
        else if (nfa == 2) 
            state2(s[i]); 
        else if (nfa == 3) 
            state3(s[i]); 
    
    if (nfa == 1) { 
        return true
    
    else
        nfa = 4; 
    
    return false;
   
// Function to check for 3 b's 
static bool checkB(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 4) 
            state4(s[i]); 
        else if (nfa == 5) 
            state5(s[i]); 
        else if (nfa == 6) 
            state6(s[i]); 
    
    if (nfa == 4) { 
   
        return true
    
    else
        nfa = 7; 
    
    return false;
   
// Function to check for 3 c's 
static bool checkC(String s, int x) 
    for (int i = 0; i < x; i++) { 
        if (nfa == 7) 
            state7(s[i]); 
        else if (nfa == 8) 
            state8(s[i]); 
        else if (nfa == 9) 
            state9(s[i]); 
    
    if (nfa == 7) { 
   
        return true
    
    return false;
   
// Driver Code 
public static void Main(String[] args)
    String s = "bbbca"
    int x = 5; 
   
    // If any of the states is true, that is, if either 
    // the number of a's or number of b's or number of c's 
    // is a multiple of three, then the string is accepted 
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) { 
        Console.WriteLine("ACCEPTED"); 
    
   
    else
        if (flag == 0) { 
            Console.WriteLine("NOT ACCEPTED"); 
               
        
        else
            Console.WriteLine("INPUT OUT OF DICTIONARY."); 
               
        
    
}
  
// This code is contributed by 29AjayKumar

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Output:

ACCEPTED

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