Alternatively, a regular language can be defined as a language recognised by a
finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem[3] (after American mathematician
Stephen Cole Kleene). In the
Chomsky hierarchy, regular languages are the languages generated by
Type-3 grammars.
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 is a regular language, A* (
Kleene star) is a regular language. Due to this, the empty string language {ε} is also regular.
If A and B are regular languages, then A ∪ B (union) and A • B (concatenation) are regular languages.
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 a's, or the language consisting of all strings of the form: several a's followed by several b's.
A simple example of a language that is not regular is the set of strings {anbn | n ≥ 0}.[4] 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.
Equivalent formalisms
A regular language satisfies the following equivalent properties:
it is the language of a regular expression (by the above definition)
Properties 10. and 11. are purely algebraic approaches to define regular languages; a similar set of statements can be formulated for a monoid M ⊆ Σ*. In this case, equivalence over M leads to the concept of a recognizable language.
Some authors use one of the above properties different from "1." as an alternative definition of regular languages.
Some of the equivalences above, particularly those among the first four formalisms, are called Kleene's theorem in textbooks. Precisely which one (or which subset) is called such varies between authors. One textbook calls the equivalence of regular expressions and NFAs ("1." and "2." above) "Kleene's theorem".[6] Another textbook calls the equivalence of regular expressions and DFAs ("1." and "3." above) "Kleene's theorem".[7] Two other textbooks first prove the expressive equivalence of NFAs and DFAs ("2." and "3.") and then state "Kleene's theorem" as the equivalence between regular expressions and finite automata (the latter said to describe "recognizable languages").[2][8] A linguistically oriented text first equates regular grammars ("4." above) with DFAs and NFAs, calls the languages generated by (any of) these "regular", after which it introduces regular expressions which it terms to describe "rational languages", and finally states "Kleene's theorem" as the coincidence of regular and rational languages.[9] Other authors simply define "rational expression" and "regular expressions" as synonymous and do the same with "rational languages" and "regular languages".[1][2]
Apparently, the term "regular" originates from a 1951 technical report where Kleene introduced "regular events" and explicitly welcomed "any suggestions as to a more descriptive term".[10]Noam Chomsky, in his 1959 seminal article, used the term "regular" in a different meaning at first (referring to what is called "
Chomsky normal form" today),[11] but noticed that his "finite state languages" were equivalent to Kleene's "regular events".[12]
Closure properties
The regular languages are
closed under various operations, that is, if the languages K and L are regular, so is the result of the following operations:
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
quotientK / 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.[15]
the reverse (or mirror image) LR.[16] Given a nondeterministic finite automaton to recognize L, an automaton for LR can be obtained by reversing all transitions and interchanging starting and finishing states. This may result in multiple starting states; ε-transitions can be used to join them.
Decidability properties
Given two deterministic finite automata A and B, it is decidable whether they accept the same language.[17]
As a consequence, using the
above closure properties, the following problems are also decidable for arbitrarily given deterministic finite automata A and B, with accepted languages LA and LB, respectively:
For regular expressions, the universality problem is
NP-complete already for a singleton alphabet.[18]
For larger alphabets, that problem is
PSPACE-complete.[19] If regular expressions are extended to allow also a squaring operator, with "A2" denoting the same as "AA", still just regular languages can be described, but the universality problem has an exponential space lower bound,[20][21][22] and is in fact complete for exponential space with respect to polynomial-time reduction.[23]
For a fixed finite alphabet, the theory of the set of all languages — together with strings, membership of a string in a language, and for each character, a function to append the character to a string (and no other operations) — is decidable, and its minimal
elementary substructure consists precisely of regular languages. For a binary alphabet, the theory is called
S2S.[24]
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.[25] On the other hand, REGULAR does not contain AC0, because the nonregular language of
palindromes, or the nonregular language can both be recognized in AC0.[26]
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).[27] 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.
Location in the 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 is not regular, one often uses the
Myhill–Nerode theorem and the
pumping lemma. Other approaches include using the
closure properties of regular languages[28] or quantifying
Kolmogorov complexity.[29]
Important subclasses of regular languages include
Finite languages, those containing only a finite number of words.[30] 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 (see
algebra of sets) including
complementation but not the
Kleene star: this class includes all finite languages.[31]
Thus, non-regularity of certain languages can be proved by counting the words of a given length in
. Consider, for example, the
Dyck language of strings of balanced parentheses. The number of words of length
in the Dyck language is equal to the
Catalan number, which is not of the form ,
witnessing the non-regularity of the Dyck language. Care must be taken since some of the eigenvalues could have the same magnitude. For example, the number of words of length in the language of all even binary words is not of the form , but the number of words of even or odd length are of this form; the corresponding eigenvalues are . In general, for every regular language there exists a constant such that for all , the number of words of length is asymptotically .[37]
The zeta function of a regular language is not in general rational, but that of an arbitrary
cyclic language is.[38][39]
Generalizations
The notion of a regular language has been generalized to infinite words (see
ω-automata) and to trees (see
tree automaton).
Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily
free. Likewise, the notion of a recognizable language (by a finite automaton) has namesake as
recognizable set over a monoid that is not necessarily free. Howard Straubing notes in relation to these facts that “The term "regular language" is a bit unfortunate. Papers influenced by
Eilenberg's monograph[40] often use either the term "recognizable language", which refers to the behavior of automata, or "rational language", which refers to important analogies between regular expressions and rational
power series. (In fact, Eilenberg defines rational and recognizable subsets of arbitrary monoids; the two notions do not, in general, coincide.) This terminology, while better motivated, never really caught on, and "regular language" is used almost universally.”[41]
^Sheng Yu (1997).
"Regular languages". In Grzegorz Rozenberg; Arto Salomaa (eds.). Handbook of Formal Languages: Volume 1. Word, Language, Grammar. Springer. p. 41.
ISBN978-3-540-60420-4.
^Eilenberg (1974), p. 16 (Example II, 2.8) and p. 25 (Example II, 5.2).
^M. Weyer: Chapter 12 - Decidability of S1S and S2S, p. 219, Theorem 12.26. In: Erich Grädel, Wolfgang Thomas, Thomas Wilke (Eds.): Automata, Logics, and Infinite Games: A Guide to Current Research.
Lecture Notes in Computer Science 2500, Springer 2002.
^Fellows, Michael R.;
Langston, Michael A. (1991). "Constructivity issues in graph algorithms". In Myers, J. Paul Jr.; O'Donnell, Michael J. (eds.). Constructivity in Computer Science, Summer Symposium, San Antonio, Texas, USA, June 19-22, Proceedings. Lecture Notes in Computer Science. Vol. 613. Springer. pp. 150–158.
doi:
10.1007/BFB0021088.
^Hopcroft, Ullman (1979), Chapter 3, Exercise 3.4g, p. 72
^Hopcroft, Ullman (1979), Theorem 3.8, p.64; see also Theorem 3.10, p.67
^Cook, Stephen; Nguyen, Phuong (2010). Logical foundations of proof complexity (1. publ. ed.). Ithaca, NY: Association for Symbolic Logic. p. 75.
ISBN978-0-521-51729-4.
^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.
^A finite language shouldn't be confused with a (usually infinite) language generated by a finite automaton.
^Volker Diekert; Paul Gastin (2008).
"First-order definable languages"(PDF). In Jörg Flum; Erich Grädel; Thomas Wilke (eds.). Logic and automata: history and perspectives. Amsterdam University Press.
ISBN978-90-5356-576-6.
^Samuel Eilenberg. Automata, languages, and machines. Academic Press. in two volumes "A" (1974,
ISBN9780080873749) and "B" (1976,
ISBN9780080873756), the latter with two chapters by Bret Tilson.
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956); it is a slightly modified version of his 1951
RAND Corporation report of the same title,
RM704.
Sakarovitch, J (1987). "Kleene's theorem revisited". Trends, Techniques, and Problems in Theoretical Computer Science. Lecture Notes in Computer Science. Vol. 1987. pp. 39–50.
doi:
10.1007/3540185356_29.
ISBN978-3-540-18535-2.
Each category of languages, except those marked by a *, is a
proper subset of the category directly above it.Any language in each category is generated by a grammar and by an automaton in the category in the same line.