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1.{<M>| M is a TM accepts any string starting with 1}

2.{<M>| M is TM accept exactly 20 strings}

Please guide

I don’t know how to apply rice theorem.

for 1. Is Tyes = { string starting with 1} Tno = { all strings – strings starting with 1}

- what is Tyes and Tno here? I only conclude by intution that when we provide strings as input some got into loop and some got accepts .

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@Learner_jai the option 2 is decidable as finite number of strings can be accepted by DFA

for option 1. its also decidable since all the strings starting with 1 , can be given by regex 1(0+1)*.

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The question is about acceptance of strings by TM. Isn't it a membership problem which is undecidable?

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@ Learner_jai (1)Tyes = { string starting with 1} Tno = { all strings – strings starting with 1} .yes these statements are right ,and we can conclude that this is non trivial property so this is undecidable.

(2) $L_{yes}=${accept exactly 20 strings(0,1,11,111....) }

$L_{no}=${0}

so we can clearly see that these two RE set in which first one satisfy the property and second one doesn't ,so we can conclude that this is non trivial property so this is undecidable.