Let $A$ be a pushdown automata with input alphabet $\Sigma$ and stack alphabet $\Gamma$ and let $R \subseteq \Gamma^∗$ be a regular language. Let $L_R(A) \subseteq \Sigma^∗$ be a language of such words over alphabet $\Sigma$, for which there exists some run of automata $A$, such as in every step symbols on the stack belongs to the regular language $R$.

Show that language $L_R(A)$ is context-free.

  • $\begingroup$ I would try to solve this using a sort-of product construction between $A$ and $R$. I'm not sure if it will suffices, but it's worth a try, I think. (Also, you should show some effort by posting your attempt.) $\endgroup$ – chi Jun 15 '18 at 10:20

Let $Q$ be the set of states in a DFA $D$ for $R$. We replace $\Gamma$ with $\Gamma \times Q$, and update the PDA so that the stack contains, apart from the original stack symbols, also the state that $D$ is in after reading the current word on the stack. The PDA only allows transitions if the element at the top of the stack corresponds to an accepting state of $D$, and also checks this condition at the very end.

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