Given a binary string S, the task is to write a program for DFA Machine that accepts a set of all strings over w ∈ (a, b)* which contains “aba” as a substring.
Examples :
Input-1 : ababa Output : Accepted Explanation : "ababa" consists "aba" Input-2 : abbbb Output : Not accepted Explanation : "abbbb" does not consist "aba"
Approach : Below is the designed DFA Machine for the given problem. Construct a transition table for DFA states and analyze the transitions between each state. Below are the steps –
Desired Language:
L = {aba, baba, abab, aababbb.....}
Explanation:
- Firstly, there will be 4states.(say q0, q1, q2, q3), withq0 being initial state and q3 being final state.
- Initially we will be in q0 state, now we start reading the given string.
- When we read ‘b’, we will remain in the same state
- If we read ‘a’, then it transits to state q1.
3. Assuming that now we are in q1 state.
- When we read ‘a’, we will remain in the same state.
- If we read ‘b’, we will transits to state q2.
4. Assuming that now we are in q2 state.
- If we read ‘a’, it transits to state q3.
- If we read ‘b’, it transits to state q0.
5. Assuming we are in final state(q3)
- We remain in the same state, when we read ‘a’ or ‘b’.
6. All strings which are accepted by this DFA will have “aba” as its substring.
Transition Table :
Current state | Final state | |
---|---|---|
a | b | |
q0 | q1 | q0 |
q1 | q1 | q2 |
q2 | q3 | q0 |
q3 | q3 | q3 |
Below is the implementation of the above approach as follows:
// C++ program for the above approach #include <stdio.h> #include <string.h> // Function to check whether the given // string is accepted by DFA or not void checkValidDFA( char s[] )
{ // Stores initial state of DFA
int initial_state = 0;
// Stores previous state of DFA
int previous_state = initial_state;
// Stores final state of DFA
int final_state;
// Iterate through the string
for ( int i = 0; i < strlen (s); i++)
{
// Checking for all combinations
if ((previous_state == 0 && s[i] == 'a' ) ||
(previous_state == 1 && s[i] == 'a' ))
{
final_state = 1;
}
if ((previous_state == 0 && s[i] == 'b' ) ||
(previous_state == 2 && s[i] == 'b' ))
{
final_state = 0;
}
if (previous_state == 1 && s[i] == 'b' )
{
final_state = 2;
}
if ((previous_state == 2 && s[i] == 'a' ) ||
(previous_state == 3))
{
final_state = 3;
}
// Update the previous_state
previous_state = final_state;
}
// If final state is reached
if (final_state == 3)
{
printf ( "Accepted" );
}
// Otherwise
else
{
printf ( "Not Accepted" );
}
} // Driver Code int main()
{ // Given string
char s[] = "ababa" ;
// Function Call
checkValidDFA(s);
} |
// C++ program for the above approach #include <cstring> #include <iostream> using namespace std;
// Function to check whether the given // string is accepted by DFA or not void checkValidDFA(string s)
{ // Stores initial state of DFA
int initial_state = 0;
// Stores previous state of DFA
int previous_state = initial_state;
// Stores final state of DFA
int final_state;
// Iterate through the string
for ( int i = 0; i < s.length(); i++) {
// Checking for all combinations
if ((previous_state == 0 && s[i] == 'a' )
|| (previous_state == 1 && s[i] == 'a' )) {
final_state = 1;
}
if ((previous_state == 0 && s[i] == 'b' )
|| (previous_state == 2 && s[i] == 'b' )) {
final_state = 0;
}
if (previous_state == 1 && s[i] == 'b' ) {
final_state = 2;
}
if ((previous_state == 2 && s[i] == 'a' )
|| (previous_state == 3)) {
final_state = 3;
}
// Update the previous_state
previous_state = final_state;
}
// If final state is reached
if (final_state == 3) {
cout << "Accepted" << endl;
}
// Otherwise
else {
cout << "Not Accepted" << endl;
}
} // Driver Code int main()
{ // Given string
string s = "ababa" ;
// Function Call
checkValidDFA(s);
} |
import java.util.*;
public class Main {
// Function to check whether the given
// string is accepted by DFA or not
public static void checkValidDFA(String s) {
// Stores initial state of DFA
int initial_state = 0 ;
// Stores previous state of DFA
int previous_state = initial_state;
// Stores final state of DFA
int final_state = 0 ;
// Iterate through the string
for ( int i = 0 ; i < s.length(); i++) {
// Checking for all combinations
if ((previous_state == 0 && s.charAt(i) == 'a' ) ||
(previous_state == 1 && s.charAt(i) == 'a' )) {
final_state = 1 ;
}
if ((previous_state == 0 && s.charAt(i) == 'b' ) ||
(previous_state == 2 && s.charAt(i) == 'b' )) {
final_state = 0 ;
}
if (previous_state == 1 && s.charAt(i) == 'b' ) {
final_state = 2 ;
}
if ((previous_state == 2 && s.charAt(i) == 'a' ) ||
(previous_state == 3 )) {
final_state = 3 ;
}
// Update the previous_state
previous_state = final_state;
}
// If final state is reached
if (final_state == 3 ) {
System.out.println( "Accepted" );
}
// Otherwise
else {
System.out.println( "Not Accepted" );
}
}
// Driver Code
public static void main(String[] args) {
// Given string
String s = "ababa" ;
// Function Call
checkValidDFA(s);
}
} |
# Function to check whether the given # string is accepted by DFA or not def checkValidDFA(s):
# Stores initial state of DFA
initial_state = 0
# Stores previous state of DFA
previous_state = initial_state
# Stores final state of DFA
final_state = None
# Iterate through the string
for i in range ( len (s)):
# Checking for all combinations
if (previous_state = = 0 and s[i] = = 'a' ) or (previous_state = = 1 and s[i] = = 'a' ):
final_state = 1
if (previous_state = = 0 and s[i] = = 'b' ) or (previous_state = = 2 and s[i] = = 'b' ):
final_state = 0
if previous_state = = 1 and s[i] = = 'b' :
final_state = 2
if (previous_state = = 2 and s[i] = = 'a' ) or (previous_state = = 3 ):
final_state = 3
# Update the previous_state
previous_state = final_state
# If final state is reached
if final_state = = 3 :
print ( "Accepted" )
# Otherwise
else :
print ( "Not Accepted" )
# Driver Code if __name__ = = '__main__' :
# Given string
s = "ababa"
# Function Call
checkValidDFA(s)
|
// C# program for the above approach using System;
class Program
{ // Function to check whether the given // string is accepted by DFA or not static void checkValidDFA( string s)
{ // Stores initial state of DFA int initial_state = 0;
// Stores previous state of DFA
int previous_state = initial_state;
// Stores final state of DFA
int final_state = 0;
// Iterate through the string
for ( int i = 0; i < s.Length; i++)
{
// Checking for all combinations
if ((previous_state == 0 && s[i] == 'a' )
|| (previous_state == 1 && s[i] == 'a' ))
{
final_state = 1;
}
if ((previous_state == 0 && s[i] == 'b' )
|| (previous_state == 2 && s[i] == 'b' ))
{
final_state = 0;
}
if (previous_state == 1 && s[i] == 'b' )
{
final_state = 2;
}
if ((previous_state == 2 && s[i] == 'a' )
|| (previous_state == 3))
{
final_state = 3;
}
// Update the previous_state
previous_state = final_state;
}
// If final state is reached
if (final_state == 3)
{
Console.WriteLine( "Accepted" );
}
// Otherwise
else
{
Console.WriteLine( "Not Accepted" );
}
} // Driver Code static void Main()
{ // Given string
string s = "ababa" ;
// Function Call
checkValidDFA(s);
} } |
// Function to check whether the given // string is accepted by DFA or not function checkValidDFA(s) {
// Stores initial state of DFA
let initial_state = 0;
// Stores previous state of DFA
let previous_state = initial_state;
// Stores final state of DFA
let final_state = null ;
// Iterate through the string
for (let i = 0; i < s.length; i++) {
// Checking for all combinations
if ((previous_state === 0 && s[i] === 'a' ) || (previous_state === 1 && s[i] === 'a' )) {
final_state = 1;
}
if ((previous_state === 0 && s[i] === 'b' ) || (previous_state === 2 && s[i] === 'b' )) {
final_state = 0;
}
if (previous_state === 1 && s[i] === 'b' ) {
final_state = 2;
}
if ((previous_state === 2 && s[i] === 'a' ) || previous_state === 3) {
final_state = 3;
}
// Update the previous_state
previous_state = final_state;
}
// If final state is reached
if (final_state === 3) {
console.log( "Accepted" );
}
// Otherwise
else {
console.log( "Not Accepted" );
}
} // Driver Code let s = "ababa" ;
checkValidDFA(s); |
Accepted
Time Complexity : O(N)
Auxiliary Space : O(1)