Dyck Word

In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of brackets. The set of Dyck words forms a Dyck language. The simplest, Dyck-1, uses just two matching brackets, e.g. ( and ).

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

Let $\Sigma = \$ be the alphabet consisting of the symbols and Let $\Sigma^$ denote its Kleene closure.
The Dyck language is defined as:
: $\.$

Context-free grammar

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 , and the production:

:

That is, ''S'' is either the empty string () 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.

Alternative definition

In yet other contexts it may instead be helpful to define the Dyck language by splitting $\Sigma^$ into equivalence classes, as follows.
For any element $u \in \Sigma^$ of length $, u ,$, we define partial functions $\operatorname : \Sigma^ \times \mathbb \rightarrow \Sigma^$ and $\operatorname : \Sigma^ \times \mathbb \rightarrow \Sigma^$ by

:$\operatorname(u, j)$ is $u$ with "$[]$" inserted into the $j$th position
:$\operatorname(u, j)$ is $u$ with "$[]$" deleted from the $j$th position

with the understanding that $\operatorname(u, j)$ is undefined for $j > , u,$ and $\operatorname(u, j)$ is undefined if $j > , u, - 2$. We define an equivalence relation $R$ on $\Sigma^$ as follows: for elements $a, b \in \Sigma^$ we have $(a, b) \in R$ if and only if there exists a sequence of zero or more applications of the $\operatorname$ and $\operatorname$ functions starting with $a$ and ending with $b$. That the sequence of zero operations is allowed accounts for the reflexivity of $R$. Symmetry follows from the observation that any finite sequence of applications of $\operatorname$ to a string can be undone with a finite sequence of applications of $\operatorname$. Transitivity is clear from the definition.

The equivalence relation partitions the language $\Sigma^$ into equivalence classes. If we take $\epsilon$ to denote the empty string, then the language corresponding to the equivalence class $\operatorname(\epsilon)$ is called the Dyck language.

Generalizations

Typed Dyck language

There exist variants of the Dyck language with multiple delimiters, e.g., Dyck-2 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. (The counting algorithm above does not generalise).
For example, the following is a valid sentence in Dyck-3 (with matching delimiters colored the same):
:

Finite depth

A Dyck language sentence can be pictured as a descent and ascent through the levels of nested brackets. As one reads along a Dyck sentence, each opening bracket increases the nesting depth by 1, and each closing bracket decreases by 1. The depth of a sentence is the maximal depth reached within the sentence.

For example, we can annotate the following sentence with the depth at each step:

0 ( 1 2 [ 3 2 2 ">3_.html" ;"title="2 [ 3 ">2 [ 3 2 2 1 ( 2 ) 1 1 ) 0 [ 1 ">3_">2_[_3_<_a>2__2_.html" ;"title="3_.html" ;"title="2 [ 3 ">2 [ 3 2 2 ">3_.html" ;"title="2 [ 3 ">2 [ 3 2 2 1 ( 2 ) 1 1 ) 0 [ 1 0

and the entire sentence has depth 3.

We define Dyck-(k, m) as the language with k types of brackets and maximal depth m. This has applications in the formal theory of Recurrent neural network">recurrent neural networks