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Intermediate Topic 21 of 51 Theory & Discrete Math

Regular Languages

Languages recognized by finite automata.

How it works

Regular languages are those that can be recognized by finite automata, described by regular expressions or generated by regular grammars; these three views are equivalent, by Kleene's theorem. The class is closed under union, intersection, complement, concatenation, Kleene star and reversal. Decision questions such as emptiness, membership and equivalence are decidable. The Myhill-Nerode theorem characterizes them by the number of distinguishable string classes.

Step-by-step walkthrough

  1. Many descriptions give the same class of languages.
  2. A DFA recognizes them.
  3. An NFA recognizes them.
  4. A regular expression describes them.
  5. The class is closed under union, concatenation, star and complement.

Where it is used

Lexical analysis in compilers, input validation, search patterns, network packet filters and log parsing use regular languages.

Common pitfall

Many natural languages and structures are not regular, such as balanced brackets, palindromes and a to the n b to the n, and need stronger models. Practical regex features such as backreferences go beyond regular languages. To show a language is not regular, use the pumping lemma or Myhill-Nerode; failing to build a DFA is not proof. Be careful with closure properties.

Where it fits in the course

Regular Languages is topic 21 of 51 in Theory & Discrete Math, rated Intermediate. It follows Nondeterministic Finite Automata. The next topic is Regular Expressions in Theory.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Theory & Discrete Math quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Regular Languages?
Languages recognized by finite automata. It belongs to Theory & Discrete Math and is an intermediate-level topic.
Where do people use Regular Languages?
Lexical analysis in compilers, input validation, search patterns, network packet filters and log parsing use regular languages.
What is the most common mistake with Regular Languages?
Many natural languages and structures are not regular, such as balanced brackets, palindromes and a to the n b to the n, and need stronger models.
What should I learn before this topic?
Start with Nondeterministic Finite Automata, the topic just before it in Theory & Discrete Math.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Theory & Discrete Math quiz.
Is Regular Languages hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Regular Languages?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Regular Languages?
Nondeterministic Finite Automata, Regular Expressions in Theory, Deterministic Finite Automata are closely connected. The full list is in the related topics section of this page.
What should I study after Regular Languages?
Continue with Regular Expressions in Theory, the next topic in Theory & Discrete Math.

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