Verifying whether One-Tape Turing Machines Run in Linear Time

We discuss the following family of problems, parameterized by integersC ≥ 2 andD ≥ 1: Does a given one-tape q-state Turing machine make at mostCn+D steps on all computations on all inputs of length n, for all n? Assuming a fixed tape and input alphabet, we show that these problems are co-NP-complete and we provide good lower bounds. Specifically, these problems can not be solved in o(q(C−1)/4) non-deterministic time by multi-tape Turing machines. We also show that the complements of these problems can be solved in O(q) non-deterministic time and not in o(q(C−1)/4) non-deterministic time by multi-tape Turing machines.