Verifying time complexity of Turing machines

We show that, for all reasonable functions T (n) = o(n log n), we can algorithmically verify whether a given one-tape Turing machine runs in time at most T (n). This is a tight bound on the order of growth for the function T because we prove that, for T (n) ≥ (n+1) and T (n) = Ω(n log n), there exists no algorithm that would verify whether a given one-tape Turing machine runs in time at most T (n). We give results also for the case of multi-tape Turing machines. We show that we can verify whether a given multi-tape Turing machine runs in time at most T (n) iff T (n0) < n0 + 1 for some n0 ∈ N. We prove a very general undecidability result stating that, for any class of functions F that contains arbitrary large constants, we cannot verify whether a given Turing machine runs in time T (n) for some T ∈ F . In particular, we cannot verify whether a Turing machine runs in constant, polynomial or exponential time.