Regular language

In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be expressed using a regular expression, in the strict sense of the latter notion used in theoretical computer science (as opposed to many regular expressions engines provided by modern programming languages, which are augmented with features that allow recognition of languages that cannot be expressed by a classic regular expression).

Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem. In the Chomsky hierarchy, regular languages are defined to be the languages that are generated by Type-3 grammars (regular grammars).

Regular languages are very useful in input parsing and programming language design.

Formal definition

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

Omega-regular language

The ω-regular languages are a class of ω-languages which generalize the definition of regular languages to infinite words. Büchi showed in 1962 that ω-regular languages are precisely the ones definable in a particular monadic second-order logic called S2S.

Formal definition

An ω-language L is ω-regular if it has the form

The elements of A^ω are obtained by concatenating words from A infinitely many times. Note that if A is regular, A^ω is not necessarily ω-regular, since A could be {ε}, the set containing only the empty string, in which case A^ω=A, which is not an ω-language and therefore not an ω-regular language.

Equivalence to Büchi automaton

Theorem: An ω-language is recognized by a Büchi automaton iff it is an ω-regular language.