DFA for accepting the language L = { anbm | n+m=even }

Design a deterministic finite automata(DFA) for accepting the language L =
For creating DFA for language, L = { a^n b^m ; n+m=even } use elementary mathematics, which says- 
even + even = even and odd + odd = even

Examples:

Input:  a a b b     // n = 2, m = 2, 2 + 2 = 4 (even)
Output:  ACCEPTED

Input:  a a a b b b b      // n = 3, m = 4, 3 + 4 = 7 (odd) 
Output:  NOT ACCEPTED

Input:  a a a b b b     // n = 3, m = 3, 3 + 3 = 6 (even)
Output:  ACCEPTED

Approaches:
There is 2 cases that results in acceptance of string: 

1. If both n and m are even then their sum will be even
2. If both n and m are odd then their sum will be even

Description:
Given DFA has 2 parts. The first part consists of states 0, 1, 5, and 6 which is for both n and m is odd. The second part consists of states 2, 3, and 4 is for both n and m is even.
DFA State Transition Diagram:

Let’s see the code for the demonstration:

C++

// C++ program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }
#include <bits/stdc++.h>
using namespace std;

// dfa tells the number associated
// with the present state
int dfa = 0;

// This function is for
// the starting state (zeroth) of DFA
void start(char c)
{
    if (c == 'a')
        dfa = 1;
    else if (c == 'b')
        dfa = 2;

    // -1 is used to check for any invalid symbol
    else
        dfa = -1;
}

// This function is for the first state of DFA
void state1(char c)
{
    if (c == 'a')
        dfa = 0;
    else if (c == 'b')
        dfa = 5;
    else
        dfa = -1;
}

// This function is for the second state of DFA
void state2(char c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the third state of DFA
void state3(char c)
{
    if (c == 'b')
        dfa = 4;
    else
        dfa = -1;
}

// This function is for the fourth state of DFA
void state4(char c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the fifth state of DFA
void state5(char c)
{
    if (c == 'b')
        dfa = 6;
    else
        dfa = -1;
}

// This function is for the sixth state of DFA
void state6(char c)
{
    if (c == 'b')
        dfa = 5;
    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)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else if (dfa == 6)
            state6(str[i]);

        else
            return 0;
    }
    if (dfa == 3 || dfa == 5)
        return 1;
    else
        return 0;
}

// Driver code
int main()
{
    char str[] = "aaabbb";
    if (isAccepted(str))
        cout << "\nACCEPTED\n";
    else
        cout << "NOT ACCEPTED\n";
    return 0;
}

// This code is contributed by SHUBHAMSINGH10

C

// C program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }
#include <stdio.h>
#include <string.h>

// dfa tells the number associated
// with the present state
int dfa = 0;

// This function is for
// the starting state (zeroth) of DFA
void start(char c)
{
    if (c == 'a')
        dfa = 1;
    else if (c == 'b')
        dfa = 2;

    // -1 is used to check for any invalid symbol
    else
        dfa = -1;
}

// This function is for the first state of DFA
void state1(char c)
{
    if (c == 'a')
        dfa = 0;
    else if (c == 'b')
        dfa = 5;
    else
        dfa = -1;
}

// This function is for the second state of DFA
void state2(char c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the third state of DFA
void state3(char c)
{
    if (c == 'b')
        dfa = 4;
    else
        dfa = -1;
}

// This function is for the fourth state of DFA
void state4(char c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the fifth state of DFA
void state5(char c)
{
    if (c == 'b')
        dfa = 6;
    else
        dfa = -1;
}

// This function is for the sixth state of DFA
void state6(char c)
{
    if (c == 'b')
        dfa = 5;
    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)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else if (dfa == 6)
            state6(str[i]);

        else
            return 0;
    }
    if (dfa == 3 || dfa == 5)
        return 1;
    else
        return 0;
}

// driver code
int main()
{
    char str[] = "aaabbb";
    if (isAccepted(str))
        printf("\nACCEPTED\n");
    else
        printf("NOT ACCEPTED\n");
    return 0;
}

Java

// Java program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }
class GFG {

    // dfa tells the number associated
    // with the present state.
    static int dfa = 0;

    // This function is for
    // the starting state (Q0)of DFA
    static void start(char c)
    {
        if (c == 'a') {
            dfa = 1;
        }
        else if (c == 'b') {
            dfa = 2;
        }

        // -1 is used to check for any invalid symbol
        else {
            dfa = -1;
        }
    }

    // This function is for
    // the first state (Q1) of DFA
    static void state1(char c)
    {
        if (c == 'a') {
            dfa = 0;
        }
        else if (c == 'b') {
            dfa = 5;
        }
        else {
            dfa = -1;
        }
    }

    // This function is for the
    // second state (Q2) of DFA
    static void state2(char c)
    {
        if (c == 'b') {
            dfa = 3;
        }
        else {
            dfa = -1;
        }
    }

    // This function is for the
    // third state (Q3)of DFA
    static void state3(char c)
    {
        if (c == 'b') {
            dfa = 4;
        }
        else {
            dfa = -1;
        }
    }

    // This function is for the
    // fourth state (Q4) of DFA
    static void state4(char c)
    {
        if (c == 'b') {
            dfa = 3;
        }
        else {
            dfa = -1;
        }
    }

    // This function is for the
    // fifth state (Q5) of DFA
    static void state5(char c)
    {
        if (c == 'b') {
            dfa = 6;
        }
        else {
            dfa = -1;
        }
    }

    // This function is for the
    // sixth state (Q6) of DFA
    static void state6(char c)
    {
        if (c == 'b') {
            dfa = 5;
        }
        else {
            dfa = -1;
        }
    }

    static int isAccepted(char str[])
    {
        // store length of string
        int i, len = str.length;

        for (i = 0; i < len; i++) {
            if (dfa == 0)
                start(str[i]);

            else if (dfa == 1)
                state1(str[i]);

            else if (dfa == 2)
                state2(str[i]);

            else if (dfa == 3)
                state3(str[i]);

            else if (dfa == 4)
                state4(str[i]);

            else if (dfa == 5)
                state5(str[i]);

            else if (dfa == 6)
                state6(str[i]);

            else
                return 0;
        }
        if (dfa == 3 || dfa == 5)
            return 1;
        else
            return 0;
    }

    // Driver code
    public static void main(String[] args)
    {
        char str[] = "aaabbb".toCharArray();
        if (isAccepted(str) == 1)
            System.out.println("ACCEPTED");
        else
            System.out.println("NOT ACCEPTED");
    }
}

Python3

# Python3 program to implement DFA that accepts
# all strings which follow the language
# L = a ^ n b ^ m n + m = even

# This function is for the dfa = starting
# dfa = state (zeroth) of DFA
def start(c):
    if (c == 'a'):
        dfa = 1
    elif (c == 'b'):
        dfa = 2

    # -1 is used to check for any
    # invalid symbol
    else:
        dfa = -1
    return dfa

# This function is for the first
# dfa = state of DFA
def state1(c):
    if (c == 'a'):
        dfa = 0
    elif (c == 'b'):
        dfa = 5
    else:
        dfa = -1
    return dfa

# This function is for the second
# dfa = state of DFA
def state2(c):
    if (c == 'b'):
        dfa = 3
    else:
        dfa = -1
    return dfa

# This function is for the third
# dfa = state of DFA
def state3(c):
    if (c == 'b'):
        dfa = 4
    else:
        dfa = -1
    return dfa

# This function is for the fourth
# dfa = state of DFA
def state4(c):
    if (c == 'b'):
        dfa = 3
    else:
        dfa = -1
    return dfa

# This function is for the fifth
# dfa = state of DFA
def state5(c):
    if (c == 'b'):
        dfa = 6
    else:
        dfa = -1
    return dfa


# This function is for the sixth
# dfa = state of DFA
def state6(c):
    if (c == 'b'):
        dfa = 5
    else:
        dfa = -1
    return dfa

def isAccepted(String):

    # store length of String
    l = len(String)

    # dfa tells the number associated
    # with the present dfa = state
    dfa = 0
    for i in range(l):
        if (dfa == 0):
            dfa = start(String[i])

        elif (dfa == 1):
            dfa = state1(String[i])

        elif (dfa == 2) :
            dfa = state2(String[i])

        elif (dfa == 3) :
            dfa = state3(String[i])

        elif (dfa == 4) :
            dfa = state4(String[i])

        elif (dfa == 5) :
            dfa = state5(String[i])

        elif (dfa == 6):
            dfa = state6(String[i])

        else:
            return 0
    if(dfa == 3 or dfa == 5) :
        return 1
    else:
        return 0

# Driver code
if __name__ == "__main__" :
    String = "aaabbb"
    if (isAccepted(String)) :
        print("ACCEPTED")
    else:
        print("NOT ACCEPTED")

# This code is contributed by
# Shubham Singh(SHUBHAMSINGH10)

C#

// C# program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }
using System;

class GFG
{

    // dfa tells the number associated
    // with the present state.
    static int dfa = 0;

    // This function is for
    // the starting state (Q0)of DFA
    static void start(char c)
    {
        if (c == 'a')
        {
            dfa = 1;
        }
        else if (c == 'b')
        {
            dfa = 2;
        }

        // -1 is used to check for any invalid symbol
        else
        {
            dfa = -1;
        }
    }

    // This function is for
    // the first state (Q1) of DFA
    static void state1(char c)
    {
        if (c == 'a')
        {
            dfa = 0;
        }
        else if (c == 'b')
        {
            dfa = 5;
        }
        else
        {
            dfa = -1;
        }
    }

    // This function is for the
    // second state (Q2) of DFA
    static void state2(char c)
    {
        if (c == 'b')
        {
            dfa = 3;
        }
        else
        {
            dfa = -1;
        }
    }

    // This function is for the
    // third state (Q3)of DFA
    static void state3(char c)
    {
        if (c == 'b')
        {
            dfa = 4;
        }
        else
        {
            dfa = -1;
        }
    }

    // This function is for the
    // fourth state (Q4) of DFA
    static void state4(char c)
    {
        if (c == 'b')
        {
            dfa = 3;
        }
        else
        {
            dfa = -1;
        }
    }

    // This function is for the
    // fifth state (Q5) of DFA
    static void state5(char c)
    {
        if (c == 'b')
        {
            dfa = 6;
        }
        else
        {
            dfa = -1;
        }
    }

    // This function is for the
    // sixth state (Q6) of DFA
    static void state6(char c)
    {
        if (c == 'b')
        {
            dfa = 5;
        }
        else
        {
            dfa = -1;
        }
    }

    static int isAccepted(char []str)
    {
        // store length of string
        int i, len = str.Length;

        for (i = 0; i < len; i++)
        {
            if (dfa == 0)
                start(str[i]);

            else if (dfa == 1)
                state1(str[i]);

            else if (dfa == 2)
                state2(str[i]);

            else if (dfa == 3)
                state3(str[i]);

            else if (dfa == 4)
                state4(str[i]);

            else if (dfa == 5)
                state5(str[i]);

            else if (dfa == 6)
                state6(str[i]);

            else
                return 0;
        }
        if (dfa == 3 || dfa == 5)
            return 1;
        else
            return 0;
    }

    // Driver code
    public static void Main(String[] args)
    {
        char []str = "aaabbb".ToCharArray();
        if (isAccepted(str) == 1)
            Console.WriteLine("ACCEPTED");
        else
            Console.WriteLine("NOT ACCEPTED");
    }
}

/* This code contributed by PrinciRaj1992 */

PHP

<?php
// PHP program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }

// This function is for the
// starting state (zeroth) of DFA
function start($c, &$dfa)
{
    if ($c == 'a') $dfa = 1;
    elseif ($c == 'b') $dfa = 2;

    // -1 is used to check for any
    // invalid symbol
    else $dfa = -1;
}

// This function is for the first
// state of DFA
function state1($c, &$dfa)
{
    if ($c == 'a') $dfa = 0;
    elseif ($c == 'b') $dfa = 5;
    else $dfa = -1;
}

// This function is for the second
// state of DFA
function state2($c, &$dfa)
{
    if ($c == 'b') $dfa = 3;
    else $dfa = -1;
}

// This function is for the third
// state of DFA
function state3($c, &$dfa)
{
    if ($c == 'b') $dfa = 4;
    else $dfa = -1;
}

// This function is for the fourth
// state of DFA
function state4($c, &$dfa)
{
    if ($c == 'b') $dfa = 3;
    else $dfa = -1;
}

// This function is for the fifth
// state of DFA
function state5($c, &$dfa)
{
    if ($c == 'b') $dfa = 6;
    else $dfa = -1;
}

// This function is for the sixth
// state of DFA
function state6($c, &$dfa)
{
    if ($c == 'b') $dfa = 5;
    else $dfa = -1;
}

function isAccepted($str, &$dfa)
{

    // store length of string
    $i = 0; $len = sizeof($str);
    for ($i = 0; $i < $len; $i++)
    {
        if ($dfa == 0)
            start($str[$i], $dfa);

        elseif ($dfa == 1)
            state1($str[$i], $dfa);

        elseif ($dfa == 2)
            state2($str[$i], $dfa);

        elseif ($dfa == 3)
            state3($str[$i], $dfa);

        elseif ($dfa == 4)
            state4($str[$i], $dfa);

        elseif ($dfa == 5)
            state5($str[$i], $dfa);

        elseif ($dfa == 6)
            state6($str[$i], $dfa);

        else
            return 0;
    }

    if($dfa == 3 || $dfa == 5)
        return 1;
    else
        return 0;
}

// Driver code

// dfa tells the number associated
// with the present state
$dfa = 0;
$str = array("a", "a", "a", "b", "b", "b");

if (isAccepted($str, $dfa) != 0)
    echo "ACCEPTED";
else
    echo "NOT ACCEPTED";

// This code is contributed by
// Adesh kumar Singh(adeshsingh1)
?>

Javascript

<script>

// Javascript program to implement DFA that accepts
// all strings which follow the language
// L = { a^n b^m ; n+m=even }

// dfa tells the number associated
// with the present state
var dfa = 0;

// This function is for
// the starting state (zeroth) of DFA
function start(c)
{
    if (c == 'a')
        dfa = 1;
    else if (c == 'b')
        dfa = 2;

    // -1 is used to check for any invalid symbol
    else
        dfa = -1;
}

// This function is for the first state of DFA
function state1(c)
{
    if (c == 'a')
        dfa = 0;
    else if (c == 'b')
        dfa = 5;
    else
        dfa = -1;
}

// This function is for the second state of DFA
function state2(c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the third state of DFA
function state3(c)
{
    if (c == 'b')
        dfa = 4;
    else
        dfa = -1;
}

// This function is for the fourth state of DFA
function state4(c)
{
    if (c == 'b')
        dfa = 3;
    else
        dfa = -1;
}

// This function is for the fifth state of DFA
function state5(c)
{
    if (c == 'b')
        dfa = 6;
    else
        dfa = -1;
}

// This function is for the sixth state of DFA
function state6(c)
{
    if (c == 'b')
        dfa = 5;
    else
        dfa = -1;
}

function isAccepted(str)
{
    // Store length of string
    var i, len = (str.length);

    for (i = 0; i < len; i++) {
        if (dfa == 0)
            start(str[i]);

        else if (dfa == 1)
            state1(str[i]);

        else if (dfa == 2)
            state2(str[i]);

        else if (dfa == 3)
            state3(str[i]);

        else if (dfa == 4)
            state4(str[i]);

        else if (dfa == 5)
            state5(str[i]);

        else if (dfa == 6)
            state6(str[i]);

        else
            return 0;
    }
    if (dfa == 3 || dfa == 5)
        return 1;
    else
        return 0;
}

// Driver code
var str = "aaabbb";
if (isAccepted(str))
    document.write( "ACCEPTED");
else
    document.write( "NOT ACCEPTED");


</script>

Output:

ACCEPTED

Time Complexity: O(n) where n is the length of the given string array

Auxiliary Space: O(1)

 There is a minimal DFA for the same problem.