# A program that solves the Halting Problem for programs with N states

My question relates to the conclusions drawn from the Halting Problem. I understand that the Halting Problem proves that no program H(P,i) exists that determines if P(i) halts or not for P in general. However, I think that this does not show that there is no program HN(P,i) that computes the Halting problem for a program P with exactly N states. If we take HN to be the program that does this computation with the fewest states possible, the proof does not necessarily work:

Defining the program X(A) that halts when HN(A,A) returns "Does Not Halt" and loops forever when it returns "Halts" does not necessarily imply a contradiction. This is because when we compute HN(X, X), we only get defined behavior if X has exactly N states, which is something that must be shown to continue the proof.

In fact, it could very well be the case that there are no programs that do this computation using N states, and that HN just returns some undefined output if we were to run this simulation.

Is there something I'm missing? I am not super knowledgeable on this topic.

edit: This would not imply a flaw in the reasoning behind the Halting Problem as a counterexample, but it shows that the Halting Problem does not show that programs that solve the Halting Problem for any arbitrary program with fixed states.

• Imagine $N$ happens to be the number of states for a given universal Turing machine. Does allow you to define $H(P, i)$ in general?
– cody
Oct 24, 2023 at 22:36
• True, I did not think of this. Oct 24, 2023 at 23:29
• I think it's possible to expand the program by expanding the alphabet while keeping the number of states the same. Nov 5, 2023 at 14:47