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DFA for accepting the language L = {an bm | n+m=odd}

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  • Difficulty Level : Medium
  • Last Updated : 14 Sep, 2022
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Design a deterministic finite automata (DFA) for accepting the language L = \{a^nb^m | \text{m+n=odd}\}     For creating DFA for language L = {an bm | n+m=odd} use elementary mathematics which says,

odd + even = odd, and 
even+odd  = odd 

Examples:  

Input:  a a b b b           
Output:  ACCEPTED
// n = 2, m = 3, 2 + 3 = 5 (odd)

Input:  a a a b b b         
Output:  NOT ACCEPTED
 // n = 3, m = 3, 3 + 3 = 6 (even)

Input:  a a a a b b b       
Output:  ACCEPTED
// n = 4, m = 3, 4 + 3 = 7 (odd)

Approaches: 
There are 2 cases that result in acceptance of string: 

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

Description: 

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

DFA State Transition Diagram: 
 

 

Let’s see the code for the demonstration: 
 

C++




// C program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n+m=odd }
#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 (Q0)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 (Q1) 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 (Q2) of DFA
void state2(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else {
        dfa = -1;
    }
}
 
// This function is for the third state (Q3)of DFA
void state3(char c)
{
    if (c == 'b') {
        dfa = 4;
    }
    else {
        dfa = -1;
    }
}
 
// This function is for the fourth state (Q4) of DFA
void state4(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else {
        dfa = -1;
    }
}
 
// This function is for the fifth state (Q5) of DFA
void state5(char c)
{
    if (c == 'b') {
        dfa = 6;
    }
    else {
        dfa = -1;
    }
}
 
// This function is for the sixth state (Q6) 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 == 2 || dfa == 4 || dfa == 6 || dfa == 1)
        return 1;
    else
        return 0;
}
 
// driver code
int main()
{
    char str[] = "aaaabbbbb";
    if (isAccepted(str))
        printf("ACCEPTED");
    else
        printf("NOT ACCEPTED");
    return 0;
}

Java




// Java program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n+m=odd }
class GFG
{
 
// dfa tells the number associated
// string end in which 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 == 2 || dfa == 4 ||
        dfa == 6 || dfa == 1)
        return 1;
    else
        return 0;
}
 
// Driver code
public static void main(String[] args)
{
    char str[] = "aaaabbbbb".toCharArray();
    if (isAccepted(str) == 1)
        System.out.println("ACCEPTED");
    else
        System.out.println("NOT ACCEPTED");
}
}
 
// This code is contributed by PrinciRaj1992

Python3




# Python3 program to implement DFS that accepts 
# all Stringing which follow the language 
# L = a ^ n b ^ m n + m = odd
   
# 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 Stringing 
    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 == 1 or dfa == 2 or dfa == 4 or dfa == 6) :
        return 1
    else:
        return 0
   
# Driver code 
if __name__ == "__main__" :
    String = "aaaabbbbb"
    if (isAccepted(String)) :
        print("ACCEPTED"
    else:
        print("NOT ACCEPTED"
  

C#




// C# program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n+m=odd }
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 (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 == 2 || dfa == 4 ||
            dfa == 6 || dfa == 1)
            return 1;
        else
            return 0;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        char []str = "aaaabbbbb".ToCharArray();
        if (isAccepted(str) == 1)
            Console.WriteLine("ACCEPTED");
        else
            Console.WriteLine("NOT ACCEPTED");
    }
}
 
// This code contributed by Rajput-Ji

PHP




<?php
// PHP program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; n+m=odd }
 
// 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 == 2 || $dfa == 4 ||
       $dfa == 6 || $dfa == 1)
        return 1;
    else
        return 0;
}
 
// Driver code
 
// dfa tells the number associated
// with the present state
$dfa = 0;
$str = array("a", "a", "a", "a",   
             "b", "b", "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 DFS that accepts
// all string which follow the language
// L = [ a^n b^m ; n+m=odd }   
// dfa tells the number associated
    // string end in which state.
    var dfa = 0;
 
    // This function is for
    // the starting state (Q0)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 (Q1) 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 (Q2) of DFA
    function state2( c) {
        if (c == 'b') {
            dfa = 3;
        } else {
            dfa = -1;
        }
    }
 
    // This function is for the
    // third state (Q3)of DFA
    function state3( c) {
        if (c == 'b') {
            dfa = 4;
        } else {
            dfa = -1;
        }
    }
 
    // This function is for the
    // fourth state (Q4) of DFA
    function state4( c) {
        if (c == 'b') {
            dfa = 3;
        } else {
            dfa = -1;
        }
    }
 
    // This function is for the
    // fifth state (Q5) of DFA
    function state5( c) {
        if (c == 'b') {
            dfa = 6;
        } else {
            dfa = -1;
        }
    }
 
    // This function is for the
    // sixth state (Q6) 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 == 2 || dfa == 4 || dfa == 6 || dfa == 1)
            return 1;
        else
            return 0;
    }
 
    // Driver code
     
         var str = "aaaabbbbb";
        if (isAccepted(str) == 1)
            document.write("ACCEPTED");
        else
            document.write("NOT ACCEPTED");
 
// This code is contributed by gauravrajput1
</script>

Output:  

ACCEPTED

Time complexity: O(n) where n is the length of the given string str

Auxiliary space: O(1)


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