# “Any problem whose domain is finite is always Decidable”, is true or false?

Question no:$1$

"Any problem whose domain is finite is always Decidable"

lets take a TM,$M$ and finite domain of problem i.e. finite set of strings for eg. {$a,abaa,bba$}, Now the problem "whether $M$ accepts these set of strings or not is decidable or not?"

What I think is, this is Undecidable problem because, if $M$ loops forever any of these string then we not determine whether turing machine accept it or not.so according to this logic above statement is false.

Question no:$2$

Does a single instance of "turing machine's halting problem" is decidable?
As I given above example, I think this is also Undecidable.

• For your second question (and, by extension, your first), if the problem you intend is "Does this TM halt on this input?" then this single-instance problem is decidable. There are two deciders, one that answers "yes" and another that answers "no". We don't know which one is the right one, but that doesn't matter: a problem is decidable if there is a decider for it and in any case we have that. – Rick Decker Nov 12 '17 at 20:05

• sir,If we have to decide the membership of a string for a language then your method will work.But,lets take a turing machine $M$ and a finite language $L$={a,b,aa,bb,ab,ba} then, problem if $M$ accepts $L$ is Undecidable because of looping nature of turing machine.Then How can we say that above statement is true.If I am misunderstanding something about the above statement then please correct me. – Reena Kandari Nov 12 '17 at 11:50