MadeEasy Test Series: Theory Of Computation - Decidability

Question

L= {(M)| M is a TM and there exists an input w, |w| < 100, on which M halts}. Is L decidable?

Please explain.

Comments

My approach has been:
We can enumerate words st |w| <100.
Let there be a machine M which rejects all inputs of len > 100.
while(true):
{  i = 1
   for w in set of words of len <100:
   {
         do i steps on machine M
         if M halts on w (accept/reject), return "yes"
   }
   i = i+10
}

But here M might loop forever, because number of steps has no upper bound. What is the correct way to do this?
Answer is decidable.

Answer 1

Answer is : Not even partially Decidable i.e not even Turing Recognizable

Apply Rice Theorem.

The property TM accept Language of less than 100 strings is Non Trivial and also Non Monotonic.

Therefore it's not even Partially Decidable.

Answer 2

Testing any property of the language recognized by TM is undecidable.