Some Observations on Infinitary Complexity

Abstract

Continuing the study of complexity theory of Koepke&#39;s Ordinal Turing Machines (OTMs) that was started by Rin, Löwe and the author, we prove the following results: (1) An analogue of Ladner&#39;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 $α<β$ such that $α$ is the halting time of some OTM-program, there are decision problems that are OTM-decidable in time bounded by $|w|^β\cdotγ$ for some $γ\in\text{On}$, but not in time bounded by $|w|^α\cdotγ$ for any $γ\in\text{On}$.