Regular language

For natural language that is regulated, see List of language regulators.

In theoretical computer science and formal language theory, a regular language is a formal language that can be expressed using a regular expression. Note that the "regular expression" features provided with many programming languages are augmented with features that make them capable of recognizing languages that can not be expressed by the formal regular expressions (as formally defined below).

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:

• 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 $\{a^nb^n\,\vert\; n\ge 0\}$. 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.

Equivalence to other formalisms

A regular language satisfies the following equivalent properties:

• it is the language accepted by a nondeterministic finite automaton
• it is the language accepted by a deterministic finite automaton
• it is the language accepted by an alternating finite automaton
• 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 (Büchi-Elgot-Trakhtenbrot theorem)
• it is recognized by some finite monoid, meaning it is the preimage of a subset of a finite monoid under a homomorphism from the free monoid on its alphabet (see Myhill–Nerode theorem).

The above properties are sometimes used as alternative definition of regular languages.

Closure properties

The regular languages are closed under the various operations, that is, if the languages K and L are regular, so is the result of the following operations:

• the set theoretic Boolean operations: union $K \cup L$, intersection $K \cap L$, and complement $\bar{L}$. From this also relative complement $K-L$ follows.
• the regular operations: union $K \cup L$, concatenation $K\circ L$, and Kleene star $L^*$.
• the trio operations: string homomorphism, inverse string homomorphism, and intersection with regular languages. As a consequence they are closed under arbitrary finite state transductions, like quotient $K / L$ with a regular language. Even more, regular languages are closed under quotients with arbitrary languages: If L is regular then L/K is regular for any K.
• the reverse (or mirror image) $L^R$.

Deciding whether a language is regular

Regular language in classes of Chomsky hierarchy.

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 often uses the Myhill–Nerode theorem or the pumping lemma among other methods.

There are two purely algebraic approaches to define regular languages. If:

• Σ is a finite alphabet,
• 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,
• S is a subset of M

then the set $\{ w \in \Sigma^* \, | \, f(w) \in S \}$ is regular. Every regular language arises in this fashion.

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

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

The language L is regular if and only if the number of equivalence classes of ~ is finite (A proof of this is provided in the article on the syntactic monoid). When a language is regular, then the number of equivalence classes is equal to the number of states of the minimal deterministic finite automaton accepting L.

A similar set of statements can be formulated for a monoid $M\subset\Sigma^*$. In this case, equivalence over M leads to the concept of a recognizable language.

Complexity results

In computational complexity theory, the complexity class of all regular languages is sometimes referred to as REGULAR or REG and equals DSPACE(O(1)), the decision problems that can be solved in constant space (the space used is independent of the input size). REGULAR ≠ AC⁰, since it (trivially) contains the parity problem of determining whether the number of 1 bits in the input is even or odd and this problem is not in AC⁰. On the other hand, REGULAR does not contain AC⁰, because the nonregular language of palindromes, or the nonregular language $\{0^n 1^n : n \in \mathbb N\}$ can both be recognized in AC⁰.

If a language is not regular, it requires a machine with at least Ω(log log n) space to recognize (where n is the input size). In other words, DSPACE(o(log log n)) equals the class of regular languages. In practice, most nonregular problems are solved by machines taking at least logarithmic space.

Subclasses

Important subclasses of regular languages include

• Finite languages - those containing only a finite number of words. These are regular languages, as one can create a regular expression that is the union of every word in the language.
• Star-free languages, those that can be described by a regular expression constructed from the empty symbol, letters, concatenation and all boolean operators including complementation but not the Kleene star: this class includes all finite languages.
• Cyclic languages, satisfying the conditions $uv \in L \Leftrightarrow vu \in L$ and $w \in L \Leftrightarrow w^n \in L$.

The number of words in a regular language

Let $s_L(n)$ denote the number of words of length $n$ in $L$. The ordinary generating function for L is the formal power series

$S_L(z) = \sum_{n \ge 0} s_L(n) z^n \ .$

The generating function of a language L is a rational function if L is regular. Hence for any regular language $L$ there exist an integer constant $n_0$, complex constants $\lambda_1,\,\ldots,\,\lambda_k$ and complex polynomials $p_1(x),\,\ldots,\,p_k(x)$ such that for every $n \geq n_0$ the number $s_L(n)$ of words of length $n$ in $L$ is $s_L(n)=p_1(n)\lambda_1^n+\dotsb+p_k(n)\lambda_k^n$.

Thus, non-regularity of certain languages $L'$ can be proved by counting the words of a given length in $L'$. Consider, for example, the Dyck language of strings of balanced parentheses. The number of words of length $2n$ in the Dyck language is equal to the Catalan number $C_n\sim\frac{4^n}{n^{3/2}\sqrt{\pi}}$, which is not of the form $p(n)\lambda^n$, witnessing the non-regularity of the Dyck language. Care must be taken since some of the eigenvalues $\lambda_i$ could have the same magnitude. For example, the number of words of length $n$ in the language of all even binary words is not of the form $p(n)\lambda^n$, but the number of words of even or odd length are of this form; the corresponding eigenvalues are $2,-2$. In general, for every regular language there exists a constant $d$ such that for all $a$, the number of words of length $dm+a$ is asymptotically $C_a m^{p_a} \lambda_a^m$.

The zeta function of a language L is

$\zeta_L(z) = \exp \left({ \sum_{n \ge 0} s_L(n) \frac{z^n}{n} }\right) \ .$

The zeta function of a regular language is not in general rational, but that of a cyclic language is.

Generalizations

The notion of a regular language has been generalized to infinite words (see ω-automata) and to trees (see tree automaton).

See also

• Pumping lemma for regular languages
• Union of two regular languages

References

• Eilenberg, Samuel (1974). Automata, Languages, and Machines. Volume A. Pure and Applied Mathematics 58. New York: Academic Press. Zbl 0317.94045. 
• Salomaa, Arto (1981). Jewels of Formal Language Theory. Pitman Publishing. ISBN 0-273-08522-0. Zbl 0487.68064. 
• Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Zbl 1169.68300.  Chapter 1: Regular Languages, pp. 31–90. Subsection "Decidable Problems Concerning Regular Languages" of section 4.1: Decidable Languages, pp. 152–155.
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics: Symbolic Combinatorics. Online book, 2002.

External links

• Complexity Zoo: Class REG