On the Complexity of Learning Minimum Time-Bounded Turing Machines

The following problems about time-bounded program-size complexity are studied: (1) for a given string x, a size bound s, and a time bound t, whether there exists a Turing machine of size less than or equal to s that prints x in t moves; (2) for two given finite sets Y and Z of strings, a size bound s, and a time bound t, whether there exists a Turing machine of size less than or equal to s that operates in time t and accepts all $y \in Y$ and rejects all $z \in Z$. These problems are fundamental in complexity theory and feasible learning theory. The complexity of these problems is apparently between P and $NP$, but appears very difficult to classify precisely. These problems are attacked by the approach of relativization. It is shown that for certain variations of the problems, they could be either polynomial-time computable or not polynomial-time computable, depending on different oracles. Furthermore, there are oracles relative to which they are not complete for $NP$ under the polynomial-time Turing reductions, but are complete for $NP$ under the strong $NP$ reductions.