Dyck language

In the theory of formal languages of computer science, mathematics, and linguistics, the Dyck language is the language consisting of balanced strings of square brackets [ and ]. It is important in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. It is named after the mathematician Walther von Dyck.

Let ${\displaystyle \Sigma =\{[,]\}}$ be the alphabet consisting of the symbols [ and ] and let ${\displaystyle \Sigma ^{*}}$ denote its Kleene closure. For any element ${\displaystyle u\in \Sigma ^{*}}$ with length ${\displaystyle |u|}$ we define partial functions ${\displaystyle \operatorname {insert} :\Sigma ^{*}\times \mathbb {N} \cup \{0\}\rightarrow \Sigma ^{*}}$ and ${\displaystyle \operatorname {delete} :\Sigma ^{*}\times \mathbb {N} \rightarrow \Sigma ^{*}}$ by

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