Speed-up Theorems in Type-2 Computation Using Oracle Turing Machines

A classic result known as the speed-up theorem in machineindependent complexity theory shows that there exist some computable functions that do not have best programs for them [2, 3]. In this paper we lift this result into type-2 computation. Although the speed-up phenomenon is essentially inherited from type-1 computation, we observe that a direct application of the original proof to our type-2 speed-up theorem is problematic because the oracle queries can interfere the speed of the programs and hence the cancellation strategy used in the original proof is no longer correct at type-2. We also argue that a type-2 analog of the operator speed-up theorem [15] does not hold, which suggests that this curious speed-up phenomenon disappears in higher-typed computation beyond type-2. The result of this paper adds one more evidence to support the general type-2 complexity theory under the framework proposed in [14, 12, 13] as a reasonable setup.