determine if a machine prints a certain string in less time than it takes to run the machine itself? Does there exist a procedure that determines if a polytime machine prints a certain string, and does so in less time than the machine itself takes to run?

Define a machine $a$ that analyzes another machine $b$, input $i$, and string $r$:

$a(b,i,r) = b(i) \text{ prints } r$ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $[T_b = O(n^k), k &gt;= 2]$

Is there an $a$ with run-time $T_a \in o(T_b)$ on all $b,i,r$?
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.