Algorithmic Randomness for Infinite Time Register Machines

A concept of randomness for infinite time register machines (ITRMs), resembling Martin-L\"of-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen's theorem holds

Infinite computations with random oracles

We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent from ZFC for ordinal time machines (OTMs) with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider, infinite time Turing machines (ITTMs), unresetting and resetting infinite time register machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible ordinals \alpha

The Lost Melody Phenomenon

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set $\{x\}$ is computable while $x$ itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for $ITRM$s. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for $ITRM$s. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that $0^{\sharp}$ exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting $\alpha$-register machines and give some information on the question for which of these machines there are lost melodies

Optimal Results on ITRM-recognizability

Exploring further the properties of ITRM-recognizable reals, we provide a detailed analysis of recognizable reals and their distribution in G\"odels constructible universe L. In particular, we show that, for unresetting infinite time register machines, the recognizable reals coincide with the computable reals and that, for ITRMs, unrecognizables are generated at every index bigger than the first limit of admissibles. We show that a real r is recognizable iff it is $\Sigma_{1}$-definable over $L_{\omega_{\omega}^{CK,r}}$, that $r\in L_{\omega_{\omega}^{CK,r}}$ for every recognizable real $r$ and that either all or no real generated over an index stage $L_{\gamma}$ are recognizable