DFA accepting all strings over w ∈(a,b)* which contains “aba” as a substring

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 :

1. Firstly, there will be 4states.(say q₀, q₁, q₂, q₃) , with q₀ being initial state and q₃ being final state.
2. 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 q_0.

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 –

C++

// 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);
}

C

// 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);
}

Output :

Accepted

Time Complexity : O(N) 
Auxiliary Space : O(1)