Computable bound on halting time ($t_M(n)\le f(n)$)

A machine's halting times admit a total computable function as a bound on all inputs in its domain if and only if that domain is a computable set. A bound decides the domain by finite simulation; conversely, a domain decider allows returning zero outside the domain and the exactly simulated halting time inside it. In particular every machine computing a total computable function has an exact computable running-time function, while nontotal machines may or may not have total bounds.