We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of "supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Godel's constructible hierarchy: namely that of L where is the supremum of halting times. A number of other open questions are thereby answered. 1 Introduction: Infinite Time Turing Machines Hamkins and Lewis in [4] give an account of the construction of these machines (first developed by Kidder and Hamkins in 1989) and develop the basic theory of this notion of computability. The reader should refer to this paper for a clear and full account of their basic properties, from which all the results and definitions of this introduction are taken. But to summarise: an infinite time Turing machine has...

Abstract. 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 infinitary computation are not equivalent. Surprisingly, the class of one-tape computable functions is not closed under composition; but closing it under composition yields the full class of all infinite time computable functions. Finally, every ordinal which is clockable by an infinite time Turing machine is clockable by a one-tape machine, except certain isolated ordinals that end gaps in the clockable ordinals. Infinite time Turing machines, introduced in [HamLew∞a], extend the usual operation of Turing machines into transfinite ordinal time. By doing so, they provide a model for supertask computations, computations involving infinitely many steps, and set the stage for a mathematical analysis of what is possible in principle to achieve via supertasks. For example, it is easy to see that every arithmetic set is decidable by such machines; a bit more sophistication reveals that every Π 1 1 set and more is supertask decidable. A rich degree structure has emerged on the class of reals and sets of reals, stratified by two natural jump operators. For this and more analysis we refer the reader to the small but rapidly growing body of literature on the subject: [HamLew∞a], [HamLew∞b], [Wel∞a], [Wel∞b] and [Wel98]. Let us review how the machines work. Using a three-tape Turing machine model, with separate input, scratch and output tapes, an infinite time Turing machine progresses through the successor stages of computation just as an ordinary Turing machine does, according to the rigid instructions of a finite program running with

Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences over the ground model of the standard natural numbers:   ¢¡-Comprehension Axiom scheme) is insufficient. £-¤¦¥¨ § (second order number theory with a ©��

• We produce a classification of the pointclasses using infinite time turing machines with 1-tape. The reason for choosing this formalism is that it apparently yields a smoother classification of classes defined by algorithms that halt at limit ordinals. • We consider some relations of such classes with other similar notions, such as arithmetical quasi-inductive definitions. • It is noted that the action of ω many steps of such a machine can correspond to the double jump operator (in the usual Turing sense): a− → a ′ ′. • The ordinals beginning gaps in the “clockable ” ordinals are admissible ordinals, and the length of such gaps corresponds to the degree of reflection those ordinals enjoy. 1

Abstract. • We produce a classification of the pointclasses of sets of reals produced by infinite time turing machines with 1-tape. The reason for choosing this formalism is that it apparently yields a smoother classification of classes defined by algorithms that halt at limit ordinals. • We consider some relations of such classes with other similar notions, such as arithmetical quasi-inductive definitions. • It is noted that the action of ω many steps of such a machine can correspond to the double jump operator (in the usual Turing sense): a−→ a ′ ′. • The ordinals beginning gaps in the “clockable ” ordinals are admissible ordinals, and the length of such gaps corresponds to the degree of reflection those ordinals enjoy. 1

In this talk we consider some issues related to the Infinite Time Turing Machine (ITTM) model of Hamkins & Lewis [3]. There a standard Turing machine (with some inessential minor modifications) is allowed to run transfinitely in ordinal time. The machine’s behaviour at limit stages of time λ is completely specified by requiring that (i) the machine enter a special limit state qL; (ii) the read/write head return to the initial starting cell at the leftmost end of the tape; (iii) the cells values- which we shall assume are taken from the alphabet of {0, 1}- are the limsup of their previous values: that is if cell i on the tape has contents Ci(γ) ∈ {0, 1} at time γ, then for any i < ω Ci(λ) = lim sup γ−→λ〈Ci(γ)|γ < λ〉. The original machine specified three infinite tapes: input, scratch, and output, with a read/write head positioned over one cell from each tape simultaneously. The machine’s actions at successor stages is determined by its (finite) program in the ordinary way. A number of intriguing questions immediately spring to mind. The question of the identity of the “decidable ” reals (for which x ∈ 2 � is there a program Pe so that on input x Pe halts on input x (“Pe(x)↓”)?), and of the semi-decidable reals, is answered in Welch[5]. (Hamkins and Lewis [3] had previously showed, inter alia, that Π 1 1 predicates of reals are decidable, and that the decidable, (and semi-decidable) pointclasses of reals are strictly between Π 1 1 and ∆ 1 2 in the projective hierarchy.) We shall be concerned here rather with the question of halting times, or how long such a computation takes, if it is going to halt. Definition 1 Pe(x) ↓ α will denote that program Pe(x) ↓ in exactly α steps. Pe(x) ↓ ≤α, Pe(x) ↓ <α are defined analogously. To clarify the above: Pe(x) ↓ α means that at ordinal time α the read/write head is in particular state qs and is reading a triple of cells (one from each of the three tapes) so that it’s program determines that it go into a halting state qh. Thus a machine may halt exactly at some limit stage of time α where then qs = qL.

Abstract not found

Abstract not found

Abstract. We locate winning strategies for various Σ 0 3-games in the L-hierarchy in order to prove the following: