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Table of contents
1.
Introduction
2.
What is DFA in regular expression?
3.
State Elimination Method
3.1.
Rule 1  
3.2.
Rule 2 
3.3.
Rule 3
3.4.
Rule 4
4.
Practice Problems Based On Converting DFA TO REGULAR EXPRESSION
5.
Frequently Asked Questions
5.1.
What is meant by regular expression?
5.2.
Is every DFA a regular expression?
5.3.
Write a regular expression for a set of vowels?
5.4.
What are Regular Language and Regular Grammar?
6.
Conclusion
Last Updated: Apr 3, 2024
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DFA to Regular Expression Conversion

Author Anant Dhakad
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Master Python: Predicting weather forecasts
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Ashwin Goyal
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Introduction

In this article, we'll talk about turning DFA (that's like a machine) into regular expressions (which are patterns). It's like translating one language into another! We'll explain how this conversion works and help you understand it step by step.

DFA to Regular Expression Conversion

What is DFA in regular expression?

In regular expressions, DFA stands for Deterministic Finite Automaton. It's a theoretical model used to recognize patterns or strings in a given input. DFA processes input one character at a time and moves from one state to another based on the current character and its current state. It's used to implement pattern matching and is a fundamental concept in computer science and formal language theory.

There are two methods to convert DFA to regular expression:

  1. Arden’s method (using Arden's Lemma)
  2. State elimination method.


We will discuss the rules & steps used in the State elimination method for converting DFA to Regular Expressions (also check out some of the Examples of Regular Expressions).

Note: State elimination method can only be applied to any finite Automata (NFA, ∈-NFA, DFA etc.)

You can also read about - Simplification of CFG

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State Elimination Method

Rule 1  

There must be only one initial state, and there should not be any incoming edges to it.

If there is more than one initial state, convert all initial states to non-initial states and create a new single initial state.

If there are incoming edges to the initial state, create a new initial state with no incoming edges.

For example:

                                       Example 1

Rule 2 

There must be only one final state.

If there is more than one final state, convert all final states to non-final states and create a new single final state.

For example:

                               Example 2

Rule 3

From the final state, there should be no outgoing edges.

If there are outgoing edges from the final state, convert all final states to non-final states and create a new final state having no outgoing edges.

For example:

Example 3

Rule 4

After verifying rules 1 to 3, all intermediate states are eliminated one by one. We can eliminate these intermediate states in any order.

After following rules 1 to 4, only the initial and final states will be left. There will only be one edge going from the initial to the final state. The cost of this edge is the required regular expression.

Let us now solve an example.

Question: For the following DFA, find its equivalent regular expression?

Example 4

Step 1: We will first multiple final states into a single final state 

The resulting DFA after step 1 will be

Step 1

Step 2: Now, eliminate all the intermediary steps. After eliminating state q4, q3, q5, DFA will look like

Step 2

Further, after eliminating state q2, DFA will look like,

Step 3

Finally, we are left with only the initial & final state, and here cost of this transition is the required regular expression.

So,  Regular expression = a.(b+c+d)

Also see, Turing Machine in TOC.

Practice Problems Based On Converting DFA TO REGULAR EXPRESSION

Problem: Given a DFA (Deterministic Finite Automaton), convert it into an equivalent regular expression.

Steps for the Solution:

  1. Start by identifying the states, transitions, and accepting states of the DFA.
  2. Use the state elimination method or other techniques to systematically eliminate states while preserving the language accepted by the DFA.
  3. For each pair of states to be eliminated, derive a regular expression that represents the language accepted by the DFA before and after the elimination.
  4. Continue this process until only two states remain, representing the start and accept states of the final regular expression.
  5. Combine the regular expressions obtained from each elimination step to form the overall regular expression for the DFA.

Lets take some examples:

Problem 1: Find the regular expression for the following DFA:

problem 1

Solution: Regular Expression = 0(10)*

Problem 2: Find the regular expression for the following DFA:

problem 2

Solution: Regular Expression = c*a(d+bc*a)*

Frequently Asked Questions

What is meant by regular expression?

A regular expression can also be defined as a pattern sequence that represents a string. It is most commonly used in pattern matching with strings or string matching. 

Is every DFA a regular expression?

No, not every DFA can be directly represented as a regular expression; some DFAs have no equivalent regular expression.

Write a regular expression for a set of vowels?

The regular expression for a set of vowels is ( a ∪ e ∪ i ∪ o ∪ u ). 

What are Regular Language and Regular Grammar?

Grammar is considered regular if it has rules of the form A -> a or A -> aB or A -> ɛ where ɛ  is a special symbol known as NULL and a language is considered regular if it can be expressed using regular expressions. 

Conclusion

Cheers if you reached here!! 

In this article, we learned how to convert DFA to regular expressions and practiced an example. 

Recommended Readings


Also check out some of the Guided Paths on topics such as Data Structure and Algorithms, Competitive Programming, Interview Guide For Product Based Companies, etc. as well as some Contests, Test Series, Interview Bundles, and some Interview Experiences curated by top Industry Experts only on  Coding Ninjas Studio.

Happy Learning!!

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