Verifying whether one-tape Turing machines run in linear time

Abstract

We discuss the following family of problems, parameterized by integers C≥2 and D≥1: Does a given one-tape q-state Turing machine make at most Cn+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 cannot be solved in o(q(C−1)/4) nondeterministic time by multi-tape Turing machines. We also show that the complements of these problems can be solved in O(qC+2) nondeterministic time and cannot be solved in o(q(C−1)/4) nondeterministic time by multi-tape Turing machines.