Abstract
A classic result known as the speed-up theorem in machine-independent 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 under the notion of our type-2 complexity theory depicted in [15][13][14]. While the speed-up phenomenon is essentially inherited from type-1 computation, we cannot directly apply the original proof to our type-2 speed-up theorem 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 [16] does not hold, which suggests that this curious phenomenon disappears in higher-typed computation beyond type-2.
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Li, CC. (2007). Speed-Up Theorems in Type-2 Computation. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_50
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DOI: https://doi.org/10.1007/978-3-540-73001-9_50
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