Program to construct a DFA which accept the language L = {anbm | n mod 2=0, m≥1}

Prerequisite – Finite Automata Introduction
Problem: Design a deterministic finite automata (DFA) for accepting the language

Regular expression for above language L is,

L = (aa)*.b+

Examples:

Input:  a a b b b           
Output:  ACCEPTED
// n = 2 (even) m=3 (>=1)

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

Input:  a a a a        
Output:  NOT ACCEPTED
// n = 4, m = 0( must be >=1)

Approaches:
There are 3 steps involve which results in acceptance of string:

1. Construct FA for means having even number of a’s.
2. Construct FA for means having any number of b’s greater than one.
3. Concatenate the two FA and make single DFA.

Any other combination result is the rejection of the input string.

Description:
Given DFA has following states. State 3 leads to the acceptance of the string, whereas states 0, 1, 2 and 4 leads to the rejection of the string.

DFA State Transition Diagram:



Let’s see code for the demonstration:


C/C++
// C program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; (n)mod 2=0, m>=1 }
#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 = 3;
    }

    // -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 = 2;
    }
    else if (c == 'b') {
        dfa = 4;
    }
    else {
        dfa = -1;
    }
}

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

// This function is for the third state (Q3)of DFA
void state3(char c)
{
    if (c == 'b') {
        dfa = 3;
    }
    else if (c == 'a') {
        dfa = 4;
    }
    else {
        dfa = -1;
    }
}

// This function is for the fourth state (Q4) of DFA
void state4(char c)
{
    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
            return 0;
    }
    if (dfa == 3)
        return 1;
    else
        return 0;
}

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

// This code is contributed by SHUBHAMSINGH10.
Java
// Java program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; (n)mod 2=0, m>=1 }
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 = 3;
    }

    // -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 = 2;
    }
    else if (c == 'b')
    {
        dfa = 4;
    }
    else
    {
        dfa = -1;
    }
}

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

// This function is for the
// third state (Q3)of DFA
static void state3(char c)
{
    if (c == 'b')
    {
        dfa = 3;
    }
    else if (c == 'a')
    {
        dfa = 4;
    }
    else
    {
        dfa = -1;
    }
}

// This function is for the
// fourth state (Q4) of DFA
static void state4(char c)
{
    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
            return 0;
    }
    if (dfa == 3)
        return 1;
    else
        return 0;
}

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

// This code is contributed by 29AjayKumar
Python 3
# Python3 program to implement DFS that accepts
# all Stringing which follow the language
# L = {a ^ n b ^ m (n)mod 2 = 0, m>= 1 }

# This function is for the dfa = starting
# dfa = state (zeroth) of DFA
def start(c):
    if (c == 'a'):
        dfa = 1
    elif (c == 'b'):
        dfa = 3

    # -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 = 2
    elif (c == 'b'):
        dfa = 4
    else:
        dfa = -1
    return dfa

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

# This function is for the third
# dfa = state of DFA
def state3(c):
    if (c == 'b'):
        dfa = 3
    elif (c == 'a'):
        dfa = 4
    else:
        dfa = -1
    return dfa

# This function is for the fourth
# dfa = state of DFA
def state4(c):
    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])
        else:
            return 0
    if(dfa == 3) :
        return 1
    else:
        return 0

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

# This code is contributed by SHUBHAMSINGH10.
C#
// C# program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ; (n)mod 2=0, m>=1 }
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 = 3;
    }

    // -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 = 2;
    }
    else if (c == 'b')
    {
        dfa = 4;
    }
    else
    {
        dfa = -1;
    }
}

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

// This function is for the
// third state (Q3)of DFA
static void state3(char c)
{
    if (c == 'b')
    {
        dfa = 3;
    }
    else if (c == 'a')
    {
        dfa = 4;
    }
    else
    {
        dfa = -1;
    }
}

// This function is for the
// fourth state (Q4) of DFA
static void state4(char c)
{
    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
            return 0;
    }
    if (dfa == 3)
        return 1;
    else
        return 0;
}

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

// This code is contributed by 29AjayKumar
PHP
<?php
// PHP program to implement DFS that accepts
// all string which follow the language
// L = { a^n b^m ;(n)mod 2=0, m>=1 }

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

    elseif ($c == 'b')
        $dfa = 3;

    // -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')
        $dfa = 2;
    elseif ($c == 'b')
        $dfa = 4;
    else
        $dfa = -1;

}

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

// This function is for the third
// state of DFA
function state3($c, &$dfa)
{
    if ($c == 'b')
        $dfa = 3;
    elseif ($c == 'a')
        $dfa = 4;
    else
        $dfa = -1;
}

// This function is for the fourth
// state of DFA
function state4($c, &$dfa)
{
    $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);
        else
            return 0;

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

// Driver Code

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

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

// This code is contributed
// by Adesh Singh
?>

Output:

ACCEPTED