Some Observations on Infinitary Complexity

Abstract

Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was done in [CLR], we prove the following results:1. An analogue of Ladner's theorem for OTMs holds: That is, there are languages L which are NP ∞ , but neither P ∞ nor NP ∞complete.This answers an open question of [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 OTMdecidable in time bounded by |w| β · γ for some γ ∈ On, but not in time bounded by |w| α · γ for any γ ∈ On.