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 non-deterministic 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 non-deterministic and co-nondeterministic 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 C+2 ) non-deterministic time and not in o(q (C−1)/2 ) non-deterministic time by multi-tape Turing machines.