Formal Languages and Logic/Context-free languages

Definition (context-free grammar):

A context-free grammar is a Chomsky grammar $G=(\Sigma ,N,P,S)$ such that all productions are of the form

$V\to \alpha$,

where $V\in N$ and $\alpha \in (N\cup \Sigma )(N\cup \Sigma )^{*}$.

Definition (context-free language):

A context-free language is a language $L$ over an alphabet $\Sigma$ so that there exists a context-free Chomsky grammar $G$ that accepts precisely $L$.

Definition (Dyck language):

Let $n\in \mathbb {N}$. The Dyck language $D_{n}$ of order $n$ is the formal languange

$D_{n}:=\{w=a_{1}a_{2}\cdots a_{k}\in \Sigma ^{*}|k\in \mathbb {N} \wedge \forall (m,j)\in \{1,\ldots ,n\}\times \{1,\ldots ,k\}:|\{l\leq j|a_{l}=)_{m}\}|\leq |\{l\leq j|a_{l}=(_{m}\}|\wedge |\{l\leq n|a_{l}=)_{m}\}|=|\{l\leq n|a_{l}=)_{m}\}|\}$

whose alphabet is $\Sigma =\{(_{1},\ldots ,(_{n},)_{1},\ldots ,)_{n}\}$.