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.

Contents

Formal definition

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

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\}. This is because in this language, the number of a′s controls the number of b′s, which is not allowed by any of the above rules.

Equivalence to other formalisms

A regular language satisfies the following equivalent properties:

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

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.[1] 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.[2]

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).[3] 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.

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:

Deciding whether a language is 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 define regular languages. If:

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.

Finite languages

A specific subset within the class of regular languages is the 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.

The number of words in a regular language

For any regular language L there exist constants \lambda_1,\,\ldots,\,\lambda_k and polynomials p_1(x),\,\ldots,\,p_k(x) such that for every n the number s_L(n) of words of length n in L satisfies the equation s_L(n)=p_1(n)\lambda_1^n%2B\dotsb%2Bp_k(n)\lambda_k^n. Thus, a non-regularity of some language L' can be proved by counting the words 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.

See also

References

  1. ^ M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Systems Theory, 17:13–27, 1984.
  2. ^ Cook, Stephen; Nguyen, Phuong (2010). Logical foundations of proof complexity (1. publ. ed.). Ithaca, NY: Association for Symbolic Logic. pp. 75. ISBN 052151729X. 
  3. ^ J. Hartmanis, P. L. Lewis II, and R. E. Stearns. Hierarchies of memory-limited computations. Proceedings of the 6th Annual IEEE Symposium on Switching Circuit Theory and Logic Design, pp. 179–190. 1965.

External links

Each category of languages is a proper subset of the category directly above it. - Any automaton and any grammar in each category has an equivalent automaton or grammar in the category directly above it.