The complexity of quickly decidable ORM-decidable sets

[bibtex key=HamkinsLinetskyMiller2007:ComplexityOfQuicklyDecidableORMSets]

The Ordinal Register Machine (ORM) is one of several different machine models for infinitary computability. We classify, by complexity, the sets that can be decided quickly by ORMs. In particular, we show that the arithmetical sets are exactly those sets that can be decided by ORMs in times uniformly less than $\omega^\omega$. Further, we show that the hyperarithmetical sets are exactly those sets that can be decided by an ORM in time uniformly less than $\omega_1^{CK}$.

Post's Problem for Ordinal Register Machines

[bibtex key=HamkinsMiller2007:PostsProblemForORMs]

We study Post’s Problem for ordinal register machines, showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors earlier results for infinite-time Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.