The Recognizability Strength of Infinite Time Turing Machines with Ordinal Parameters

Abstract

Infinite time Turing machine models with tape length α, denoted Tα, 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 Tα 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:• For some ξ < α, there is a T ξ -writable but not Tα-writable subset of ω.• There is a gap in the Tα-writable ordinals.• α is uncountable in L λα . Here λα denotes the supremum of Tα-writable ordinals, i.e. those with a Tαwritable 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 → Σα, where Σα denotes the supremum of the Tα-accidentally writable ordinals.