Tree Representations via Ordinal Machines

We study sets of reals computable by ordinal Turing machines with a tape of length the ordinals that are steered by a standard Turing program. The machines halt at some ordinal time or diverge. We construct tree representations for ordinal semidecidable sets of reals from ordinal computations. The aim is to generalize uniformization results to classes of sets which are semi-decidable by computations with input-dependent bounds on the halting time. We further briefly examine the jump structure and nondeterminism.