Some Observations on Infinitary Complexity

TL;DR

This paper explores the complexity theory of Ordinal Turing Machines, establishing analogues of classical theorems, demonstrating limitations of speedup techniques, and analyzing decidability bounds related to halting times.

Contribution

It proves an analogue of Ladner's theorem for OTMs, shows the failure of the speedup theorem for OTMs, and investigates decidability bounds based on halting times.

Findings

Existence of NP-infinite but not P-infinite or NP-complete languages for OTMs.

Speedup theorem does not hold for OTMs.

Decidability bounds depend on halting times of OTMs.

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 $\gamma\in\text{On}$.