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I'm working on a theoretical computer science exercise and need some help with solving it. Here's the task:

Task: Let $C$ be a context-free language and $R$ a regular language. Show that $C \cap R$ is context-free.

My Attempt: I understand that context-free languages can be accepted by Pushdown Automata (PDA) and regular languages by Finite Automata (FA). My approach was to construct a new PDA that combines both automata, transitioning based on the operations of the PDA for $C$ and the state transitions of the FA for $R$. Specifically, the new PDA would work as follows:

  1. States: Create pairs of states $(q_C, q_R)$ where $q_C$ are states from the PDA accepting $C$ and $q_R$ are states from the FA accepting $R$.
  2. Transitions: For a transition $(q_C, a, \gamma) \rightarrow (q_C', \gamma')$ in the PDA for $C$ and a transition $(q_R, a) \rightarrow q_R'$ in the FA for $R$, the new PDA has a transition $((q_C, q_R), a, \gamma) \rightarrow ((q_C', q_R'), \gamma')$.
  3. Accepting States: A state $(q_C, q_R)$ is accepting in the new PDA if $q_C$ is an accepting state in the PDA for $C$ and $q_R$ is an accepting state in the FA for $R$.

Question: I am not sure if this approach covers all cases, or if there are any exceptions or special considerations I need to take into account. Can anyone provide feedback or suggest improvements to this method?

rzickler
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Harold
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1 Answers1

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There is one special case you must deal with. The PDA might be able to perform a $\varepsilon$-transition where the FSA can't.

Hendrik Jan
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