![Lattice of the 14 Dyck words of length 8 - <i>[</i> and <i>]</i> interpreted as <i>up</i> and <i>down</i>](https://everipedia-storage.s3.amazonaws.com/ProfilePicture/en/Dyck_language__a4a48b/Dyck_lattice_D4.svg__07921.svg){kind=avatar}
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of square brackets [ and ]. The set of Dyck words forms the Dyck language.
Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions.
Formal definition
It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal S, and the production:
That is, S is either the empty string (ε) or is "[", an element of the Dyck language, the matching "]", and an element of the Dyck language.
An alternative context-free grammar for the Dyck language is given by the production:
That is, S is zero or more occurrences of the combination of "[", an element of the Dyck language, and a matching "]", where multiple elements of the Dyck language on the right side of the production are free to differ from each other.
Properties
- The Dyck language is closed under the operation of concatenation.
- By treating as an algebraic monoid under concatenation we see that the monoid structure transfers onto the quotient , resulting in the syntactic monoid of the Dyck language. The class will be denoted .
- The syntactic monoid of the Dyck language is not commutative: if and then .
- With the notation above, but neither nor are invertible in .
- The syntactic monoid of the Dyck language is isomorphic to the bicyclic semigroup by virtue of the properties of and described above.
- By the Chomsky–Schützenberger representation theorem, any context-free language is a homomorphic image of the intersection of some regular language with a Dyck language on one or more kinds of bracket pairs.[1]
- The Dyck language with two distinct types of brackets can be recognized in the complexity class .[2]
- The number of distinct Dyck words with exactly n pairs of parentheses is the n-th Catalan number.
Examples
Generalizations
There exist variants of the Dyck language with multiple delimiters, e.g., on the alphabet "(", ")", "[", and "]". The words of such a language are the ones which are well-parenthesized for all delimiters, i.e., one can read the word from left to right, push every opening delimiter on the stack, and whenever we reach a closing delimiter then we must be able to pop the matching opening delimiter from the top of the stack.
See also