Supertask Computation

Post's problem for supertasks has both positive and negative solutions

Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post's problem, the question whether there are supertask degrees between 0 and the supertask jump 0^jump, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0^jump, but in the context of SETS of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ …

Infinite Time Turing Machines: Supertask Computation

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Symmetry for transfinite computability

Finite Turing computation has a fundamental symmetry between inputs, outputs, programs, time, and storage space. Standard models of transfinite computational break this symmetry; we consider ways to recover it and study the resulting model of computation. This model exhibits the same symmetry as finite Turing computation in universes constructible from a set of ordinals, but that statement is independent of von Neumann-Gödel-Bernays class theory.

Characterisations of Variant Transfinite Computational Models: Infinite Time Turing, Ordinal Time Turing, and Blum-Shub-Smale machines

We consider how changes in transfinite machine architecture can sometimes alter substantially their capabilities. We approach the subject by answering three open problems touching on: firstly differing halting time considerations for machines with multiple as opposed to single heads, secondly space requirements, and lastly limit rules. We: 1) use admissibility theory, $Σ_{2}$-codes and $Π_{3}$-reflection properties in the constructible hierarchy to classify the halting times of ITTMs with multiple independent heads; the same for Ordinal Turing Machines which have $\tmop{On}$ length tapes; 2) d…

Discrete Transfinite Computation

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as $α$-recursion. We conclude that, in particular, polynomial time computation on $ω$-strings is well modelled by severa…

关于$β(α)$的下界及其他$α$-ITRMs的性质

Lower bounds on $β(α)$ and other properties of $α$-ITRMs

本文研究了$α$-ITRMs的计算强度下界及其他性质，证明了某些$α$值下的下界，反驳了之前关于其强度的猜想，并展示了基数识别ITRMs的计算强度与普通ITRMs相同，以及非可解的有界停止问题、通用机器的存在和允许机器识别基数带来的计算能力提升在$α$-ITRMs中是等价的。