The Computational Complexity of the Bounded Halting Problem

Proof: If a deterministic Turing machine T decides language BHP by computing the first n steps of each of the possible computation paths of the nondeterministic Turing machine Mi applied to x, then the worst-case running-time of T must be exponential, since Mi applied to x may have a choice of an exponential number of computation paths. So if T is to decide language BHP and still run in polynomial-time, T cannot always compute the first n steps of all of the possible computation paths of Mi applied to x. But since the first n steps of all of the possible computation paths of Mi applied to x completely determine whether Mi accepts x within n steps, in order for T to decide language BHP, it is still necessary for T to have access to information about the first n steps of all of the possible computation paths of Mi applied to x, even if T does not compute this information, i.e., T must have access to precomputed information about the first n steps of all but at most a polynomial number of the possible computation paths of Mi applied to x in order for T to decide language BHP and still run in polynomial-time. And such precomputed information must be finite, since if it were infinite, then T would require an infinite oracle tape, which would contradict our original assumption that T is just an ordinary deterministic Turing machine with no access to an oracle. But even if such precomputed information were finite, it could only be used to speed up the running-time of the Turing machine T by a constant amount of time (not the exponential amount of time required for T to run in polynomial-time and still decide language BHP), since such precomputed information could have just as easily been computed by T in a constant amount of time. So precomputed information is not sufficient to speed up the running-time of T to the point where T is able to determine in polynomial-time whether Mi accepts x within n steps. Therefore, there cannot exist a deterministic Turing machine T which decides language BHP in polynomial-time.