How to Construct a Deterministic Finite Automaton (DFA) for Strings Ending with 010 Over the Alphabet Set {0,1}

In this article, we are going to explore the construction of a Deterministic Finite Automaton (DFA) that accepts any string over the alphabet set {0,1} which ends with the sequence 010. We will define the states, transitions, and the structure of the DFA, and provide a clear explanation of its operation.

Steps to Construct the DFA

Define the States

The construction of the DFA involves defining a set of states that represent different conditions based on the sequence of input characters read so far.

State q_0: The initial state, where no characters have been read. State q_1: The state after reading the first character 0. State q_2: The state after reading the sequence 01. State q_3: The accepting state, where the sequence 010 has been fully matched.

Define the Transitions

The transitions in the DFA are defined as follows:

From q_0, on reading a 0, transition to q_1, and on reading a 1, stay in q_0. From q_1, on reading a 0, stay in q_1, and on reading a 1, transition to q_2. From q_2, on reading a 0, transition to q_1, and on reading a 1, transition to q_3. From q_3, on reading a 0, transition to q_1, and on reading a 1, transition to q_0.

Define the Accepting State

The only accepting state is q_3, which signifies that the DFA has read the complete sequence 010.

DFA Diagram

Here is a graphical representation of the DFA:

DFA Diagram

The DFA is structured as follows:

  q0  -----  q1  -----  q2  -----  q3
           1        0        0        0
         v         v         v
  q0  -----  q1  -----  q2  -----
         1         0
         v
  q0  ----  q1  -----
         1

State Transitions Summary

State Input 0 Input 1 q0 q1 q0 q1 q1 q2 q2 q1 q3 q3 q1 q0

Explanation

The DFA operates by tracking the last few characters of the input string and ensuring that the sequence 010 is not broken. Once the sequence is fully matched, the DFA transitions to the accepting state q_3, and further characters are ignored unless they can start a new valid sequence.

This DFA will correctly accept any string that ends with 010.