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arXiv · 1801.10027

Some Observations on Infinitary Complexity

Merlin Carl
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Abstract

Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was started by Rin, L\"owe and the author, we prove the following results: (1) An analogue of Ladner's theorem for OTMs holds: That is, there are languages $\mathcal{L}$ which are NP$^{\infty}$, but neither P$^{\infty}$ nor NP$^{\infty}$-complete. This answers an open question of \cite{CLR}. (2) The speedup theorem for Turing machines, which allows us to bring down the computation time and space usage of a Turing machine program down by an aribtrary positive factor under relatively mild side conditions by expanding the working alphabet does not hold for OTMs. (3) We show that, for $\alpha<\beta$ such that $\alpha$ is the halting time of some OTM-program, there are decision problems that are OTM-decidable in time bounded by $|w|^{\beta}\cdot\gamma$ for some $\gamma\in\text{On}$, but not in time bounded by $|w|^{\alpha}\cdot\gamma$ for any $\gamma\in\text{On}$.

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Merlin Carl. 2018-01-30. Some Observations on Infinitary Complexity. https://doi.org/10.1007/978-3-319-94418-0_12

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