A regular language is a formal language (i.e. a possibly infinite set of finite sequences of symbols from a finite alphabet) that satisfies the following equivalent properties:

• it can be accepted by a deterministic finite state machine
• it can be accepted by a nondeterministic finite state machine
• it can be accepted by an alternating finite automaton
• it can be described by a regular expression
• it can be generated by a regular grammar
• it can be generated by a prefix grammar
• it can be accepted by a read-only Turing machine
• it can be defined in monadic second-order logic

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

• the empty language Ø is a regular language.
• the empty string language { ε } is a regular language.
• For each a ∈ Σ, the singleton language { a } is a regular language.
• If A and B are regular languages, then A U B, A • B, and A* are regular languages.
• No other languages over Σ are regular.

All finite languages are regular. Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of a's, or the language consisting of all strings of the form: several a's followed by several b's.

The results of the union, intersection and set-difference operations when applied to regular languages is itself a regular language; the complement of every regular language is a regular language as well. Reversing every string in a regular language yields another regular language. Concatenating two regular languages (in the sense of concatenating every string from the first language with every string from the second one) also yields a regular language. The shuffle operation , when applied to two regular languages, yields another regular language. The right quotient and the left quotient of a regular language by an arbitrary language is also regular.

To locate the regular languages in the Chomsky hierarchy, one notices that every regular language is context-free. The converse is not true: for example the language consisting of all strings having the same number of a's as b's is context-free but not regular. To prove that a language such as this is not regular, one uses the Myhill-Nerode theorem or the pumping lemma.

There are two purely algebraic approaches to defining regular languages. If Σ is a finite alphabet and Σ* denotes the free monoid over Σ consisting of all strings over Σ,  f : Σ* → M is a monoid homomorphism where M is a finite monoid, and S is a subset of M, then the set f ⁻¹(S) is regular. Every regular language arises in this fashion.

If L is any subset of Σ*, one defines an equivalence relation ~ on Σ* as follows: u ~ v is defined to mean

uw in L if and only if vw in L for all w in Σ*

The language L is regular if and only if the number of equivalence classes of ~ is finite; if this is the case, this number is equal to the number of states of the minimal deterministic finite automaton accepting L.

External resource

• Department of Computer Science at the University of Western Ontario: Grail+, http://www.csd.uwo.ca/research/grail/. A software package to manipulate regular expressions, finite-state machines and finite languages. Free for non-commercial use.