3b. Consider the language FIN = { | M is a Turing machine and L(M) is finite }. Prove FIN is not decidable.
Answer is as follows :
We have
FIN = { | M is a Turing machine and L(M) is finite }
So it satisfy the following properties :
So let L be a language over Turing machines. Assume that L satisfies the following properties :
For TM's
M1 and M2, if M1 ≡ M2 then M1 ∈ L ⇔ M2 ∈ L
but
There are TM's M1 and M2, such that M1 ∈ L and M2 ∉ L
Than L is undecidable because TM's for same set is not belongs to same language.
So we can say that FIN is not decidable i.e. undecidable.
if there is any query please ask in comments....
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