Reachability for infinite time Turing machines with long tapes

Abstract

Infinite time Turing machine models with tape length $α$, denoted $T_α$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $ω$. A new phenomenon is that for some countable ordinals $α$, some cells cannot be halting positions of $T_α$ given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal $δ$. We answer this by providing various characterizations. For instance, $δ$ is the least ordinal with any of the following properties: (a) For some $ξ<α$, there is a $T_ξ$-writable but not $T_α$-writable subset of $ω$. (b) There is a gap in the $T_α$-writable ordinals. (c) $α$ is uncountable in $L_{λ_α}$. Here $λ_α$ denotes the supremum of $T_α$-writable ordinals, i.e. those with a $T_α$-writable code of length $α$. We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals $λ$, $ζ$ and $Σ$, to show that $δ$ is large in the sense that it is a closure point of the function $α\mapsto Σ_α$, where $Σ_α$ denotes the supremum of the $T_α$-accidentally writable ordinals.