determine if a machine prints a certain string in less time than it takes to run the machine itself?
Your question is somewhat unclear, but here is something you can prove using the time hierarchy theorem. We will use the convention that a Turing machine computing some function prints the result at the end, and prints nothing except that. Suppose $T$ is a machine that accepts the code of a machine $P$ running in time $O(n^\alpha)$ and an input $x$, and returns the output of $P(x)$ in time $O(n^\beta)$ for $1 \leq \beta &lt; \alpha$. Using $T$, we can evaluate the function computed by $P$ in time $O(n^\beta)$, so $\mathrm{Time}(n^\alpha) \subseteq \mathrm{Time}(n^\beta)$, which contradicts the time hierarchy theorem.