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Given a context-free grammar $G$, how can one systematically construct a grammar $G_k$ such that

$$ L(G_k) = \{w \in \Sigma^* : |\text{Pref}(w) \cap L(G)| = k\} $$

where $\text{Pref}(w)$ is the set of $|w|+1$ prefixes of $w$? Assume $G$ is unambiguous, if needed.

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Unfortunately there is no such construction, as the context-free languages are not closed under this operation.

Consider the language $\{ a^nb^n \mid n\ge 1\} \cup \{ a^kb^nc^n \mid k,n\ge 1 \}$.

The language that has two prefixes in this language is $\{ a^nb^nc^n \mid n\ge 1\}\cdot \{a,b,c\}^*$, which is not context-free.

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