Dyck Language - Formal Definition

Formal Definition

Let Σ = { } be the alphabet consisting of the symbols and let Σ∗ denote its Kleene closure. For any element u ∈ Σ∗ with length |u| we define partial functions insert : Σ∗ × (N ∪ {0}) → Σ∗ and delete : Σ∗ × N → Σ∗ by

insert(u, j) = u with "" inserted into the jth position

delete(u, j) = u with "" deleted from the jth position

with the understanding that insert(u, j) is undefined for j > |u| and delete(u, j) is undefined if j > |u| − 2. We define an equivalence relation R on Σ∗ as follows: for elements a, b ∈ Σ∗ we have (a, b) ∈ R if and only if there exists a finite sequence of applications of the insert and delete functions starting with a and ending with b, where the empty sequence is allowed. That the empty sequence is allowed accounts for the reflexivity of R. Symmetry follows from the observation that any finite sequence of applications of insert to a string can be undone with a finite sequence of applications of delete. Transitivity is clear from the definition.

The equivalence relation partitions the language Σ∗ into equivalence classes. If we take ε to denote the empty string, then the language corresponding to the equivalence class Cl(ε) is called the Dyck language.