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Description

Similar to the encoding of a Turing Machine, we can encode a Push-Down Automata. Denote $\langle M \rangle$ as the encoding of PDA M, and a natural number n, is language $L = \{ \langle M \rangle | \text{ M is a PDA, L(M) contains at least 1 string w that } |w| \leq n \}$ a recursive language?

Question

By the definition of a recursive language, I have to prove that there exists a Turing Machine that can accept L and for any input string x, the TM will end up in a halt state.

But how exactly should I do to find/construct that? Or I don’t really have to construct a Turing Machine but use some existing theorem?

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1 Answer 1

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You should think in algorithms instead of thinking in Turing Machine. Turing Machines are just a very formal way to express algorithms but writing a Turing Machine even for simple algorithms is a chore.

It suffices to know that if one really wanted to, they could construct a very complicated Turing Machine to encode an algorithm.

What the question is really asking is "Is there an algorithm that, given a PDA $M$ and an integer $n$, decides if $L(M) \cap \Sigma^{\leqslant n}\neq \emptyset$?".

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