Joel David Hamkins
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.
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Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.
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 …
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every Pi^1_1 set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the Delta^1_2 sets. Our oracle concept leads to a notion of relative computability for reals and sets of reals and a rich degree str…
Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions f:R-->N, the same class of computable functions. Nevertheless, there are infinite time computable functions f:R-->R that are not one-tape computable, and so the two models of supertask computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition y…
Nondeterministic infinite time Turing machines
本文研究无限非确定性可计算性，引入非确定性无限时间图灵机，证明其可识别集合类与可判定集合类不等，即D ≠ ND ∩ coND。
We state a version of the P=?NP problem for infinite time Turing machines. It is observed that P not= NP for this version.
Infinite time Turing machines are extended in several ways to allow for iterated oracle calls. The expressive power of these machines is discussed and in some cases determined.
We propose a notion of autoreducibility for infinite time computability and explore it and its connection with a notion of randomness for infinite time machines.
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bmΔ^1_2$ in a satisfying way.
Schindler recently addressed two versions of the question P $\stackrel{?}{=}$ NP for Turing machines running in transfinite ordinal time. These versions differ in their definition of input length. The corresponding complexity classes are labelled P, NP and ${\rm P}^+,{\rm NP}^+$. Schindler showed that P $\neq$ NP and ${\rm P}^+ \neq {\rm NP}^+$. We show that P $=$ NP $\cap$ co-NP and NP $\neq$ co-NP, whereas ${\rm P}^+ \subset$ NP $\cap$ co-NP and ${\rm NP}^+ \neq $ co-NP$^+$.
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…
Extending results of Schindler [math.LO/0106087] and Hamkins and Welch [math.LO/0212046], we establish in the context of infinite time Turing machines that P is properly contained in NP intersect coNP. Furthermore, NP intersect coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P+ = (NP+ intersect coNP+) = (NP intersect coNP), though P++ is properly contained in NP++ intersect coNP++. Within any contiguous block of infinite clockable ordinals, we show that P_alpha is not equal to NP_alpha intersect coNP_alpha, but if beta begins a gap in the cloc…
Space and Time Complexity for Infinite Time Turing Machines
本文研究了无限时间 Turing 机的空间复杂度概念，回答了关于这些概念的多个开放问题，包括低空间复杂度是否意味着低时间复杂度以及 P=PSPACE、P_+=PSPACE_+ 和 P__+=PSPACE__+ 是否成立等问题，并展示了 ITTMs 空间复杂度类之间的分离结果。
Toward higher-order infinite time Turing machines: simulational $Γ$-machines
本文定义了一族满足特定约束的极限规则，构建了具有通用机器且可写序数与时钟序数上确界相匹配的高阶无限时间图灵机模型，并通过反例验证了约束的必要性。
In Mathias forcing, conditions are pairs $(D,S)$ of sets of natural numbers, in which $D$ is finite, $S$ is infinite, and $\max D < \min S$. The Turing degrees and computational characteristics of generics for this forcing in the special (but important) case where the infinite sets $S$ are computable were thoroughly explored by Cholak, Dzhafarov, Hirst, and Slaman~\cite{CDHS-2014}. In this paper, we undertake a similar investigation for the case where the sets $S$ are members of general countable Turing ideals, and give conditions under which generics for Mathias forcing over one ideal compute…
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 α-Turing machines for countable …
This note introduces a generalization to the setting of infinite-time computation of the busy beaver problem from classical computability theory, and proves some results concerning the growth rate of an associated function. In our view, these results indicate that the generalization is both natural and promising.
We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real function or a probability measure, for example. The model is based on TTE, the representation approach for computable analysis. As a main result we prove that the functions that are computable via given representations are closed under GTM programming. This generalizes the well known fact that these functions are closed under composition. The theorem allows to spe…
We introduce a model of infinitary computation which enhances the infinite time Turing machine model slightly but in a natural way by giving the machines the capability of detecting cardinal stages of computation. The computational strength with respect to ITTMs is determined to be precisely that of the strong halting problem and the nature of the new characteristic ordinals (clockable, writable, etc.) is explored.
Randomness via infinite computation and effective descriptive set theory
本文研究了超越Π¹₁随机性及马丁-洛夫类型变体的随机性概念，探讨了无限时间图灵机所定义的类严格位于Π¹₁和Σ¹₂之间，并证明了与经典随机性概念相似的性质，同时揭示了与超算术设置的差异。
We introduce infinite time computable model theory, the computable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of ZFC.
Using nonstandard analysis, we will extend the classical Turing machines into the internal Turing machines. The internal Turing machines have the capability to work with infinite ($*$-finite) number of bits while keeping the finite combinatoric structures of the classical Turing machines. We will show the following. The internal deterministic Turing machines can do in $*$-polynomial time what a classical deterministic Turing machine can do in an arbitrary finite amount of time. Given an element of $<M;x>\in HALT$ (more precisely, the $*$-embedding of $HALT$), there is an internal deterministic…
We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(Σ_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(Σ_1^1)$ is the class of 2-differences of $Σ_1^1$-sets. Secondly, it is $Σ_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.
We discuss the question of Ralf-Dieter Schindler whether for infinite time Turing machines P^f = NP^f can be true for any function f from the reals into omega_1. We show that ``almost everywhere'' the answer is negative.
Reachability for infinite time Turing machines with long tapes
本文刻画了长纸带无限时间图灵机中最小不可达序数$\delta$，证明其满足多种等价性质，并利用Welch子模型类比证明$\delta$是$\alpha\mapsto\Sigma_\alpha$的闭包点。
We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence rel…
An ω-language is a set of infinite words over a finite alphabet X. We consider the class of recursive ω-languages, i.e. the class of ω-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class Σ11 of (effective) analytic subsets of Xω for some finite alphabet X. We investigate here the notion of ambiguity for recursive ω-languages with regard to acceptance by Büchi Turing machines. We first present in detail essentials on the literature on ω-languages accepted by Turing Machines. Then we give a complete and broad view on the notion of ambiguity and unambi…
Ramsey's theorem asserts that every $k$-coloring of $[ω]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[ω]^n$ whose solutions compute the halting set. On the other hand, for every computable $k$-coloring of $[ω]^2$ and every non-computable set $C$, there is an infinite monochromatic set $H$ such that $C \not \leq_T H$. The latter property is known as cone avoidance. In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal st…
Undecidability of Finite Orbit Recognition in Polynomial Maps
研究证明判定图灵机是否产生最终周期轨迹不可判定，从而推导出多项式动力系统中无限整数元组轨道有限性问题的不可判定性。
Infinite Time Recognizability from Random Oracles and the Recognizable Jump Operator
本文研究了从随机 oracle 的无限时间可识别性，并引入了可识别性跃迁运算符，展示了 ITTMs 和 ITRMs 的可识别性跃迁在原始递归上等价，解决了可识别性度的 Post 问题。
Recognizable sets and Woodin cardinals: Computation beyond the constructible universe
本文研究了在构造宇宙外通过序数时间 Turing 机进行计算的可识别集，证明了可计算集属于 L，而无限时间计算的可识别对象可达 Woodin 数字的层次。
The distribution of ITRM-recognizable reals
本文研究了ITRM可识别实数在哥德尔构造层次L的canonical well-ordering中的分布，证明了可识别实数在L中存在空隙，并探讨了可识别性的相对化概念。
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。
Loops, Inverse Limits and Non-Determinism
本文在Weihrauch复杂度中引入无限循环（逆向极限）算子，证明其单调性及弱Kőnig引理下的封闭性，从而保证非确定性计算类在编程中的灵活性，并给出递归定理单射版与独立选择定理无限版作为工具。
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