Regular Language - Formal Definition

Formal Definition

The collection of regular languages over an alphabet Σ is defined recursively as follows:

• The empty language Ø is a regular language.
• For each a ∈ Σ (a belongs to Σ), the singleton language {a} is a regular language.
• If A and B are regular languages, then A ∪ B (union), A • B (concatenation), and A* (Kleene star) are regular languages.
• No other languages over Σ are regular.

See regular expression for its syntax and semantics. Note that the above cases are in effect the defining rules of regular expression.

Examples

All finite languages are regular; in particular the empty string language {ε} = Ø* is regular. Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of as, or the language consisting of all strings of the form: several as followed by several bs.

A simple example of a language that is not regular is the set of strings . Intuitively, it cannot be recognized with a finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a's. Techniques to prove this fact rigorously are given below.