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Membership problem of Recursive languages are decidable.

My approach:

Let $L$ be a recursive language and $M$ be the Turing Machine that accepts it. For string $w,$ if $w ∈ L,$ then $M$ halts in final state. If $w ∉ L,$ then $M$ halts in non-final state. (halts always!). That's why Recursive languages are decidable for Membership problem

My question is in same logic, why finiteness, emptiness is undecidable? Don't want any concrete proof. I just want brief concepts like my approach.

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To decide whether a language is empty you'd have to run $M$ on all possible input strings and verify that $M$ always rejects. How are you going to do that in a way that ensures that your algorithm always terminates?

Similarly, to decide whether $L$ is finite you'd have to check that $M$ rejects almost always, i.e., always except for at most a finite number of inputs.

Formally, you can reduce the halting problem to problem of deciding whether a language $L$, encoded as a decider machine for $L$, is empty/finite.

For example, given Turing machine $T$ you can define a Turing machine $T'$ that takes a string $x$ as input and simulates $T$ on empty input for $|x|$ steps. If the simulation ends, $T'$ accepts. Otherwise $T'$ rejects. Then, you can decide if $T$ halts on empty input by checking if the language recognized by $T'$ is not empty.

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  • $\begingroup$ one thing tell reduction only possible to undecidable to undecidable? Reduction Decidable to undecidable possible? $\endgroup$
    – Punia
    Oct 14 at 11:09
  • $\begingroup$ empty input means $\endgroup$
    – Punia
    Oct 14 at 12:09
  • $\begingroup$ I'm not sure how it helps, but yes: it is possible to reduce a decidable language to an undecidable one. Empty input means that the contents of the tape are the empty word, usually denoted with $\varepsilon$. $\endgroup$
    – Steven
    Oct 14 at 12:29
  • $\begingroup$ Your answer is good.. I need understand your last paragraph... $\endgroup$
    – Punia
    Oct 14 at 12:36
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    $\begingroup$ The first two paragraphs are only intuition. In mathematics, intuition is important, but only proof counts. $\endgroup$ Oct 14 at 15:50

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