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. Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP ∩co-NP. Furthermore, NP ∩co-NP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P + = NP + ∩co-NP + = NP ∩co-NP, though P ++ is properly contained in NP ++ ∩co-NP ++. Within any contiguous block of infinite clockable ordinals, we show that Pα ̸ = NPα ∩co-NPα, but if β begins a gap in the clockable ordinals, then Pβ = NPβ ∩co-NPβ. Finally, we establish that P f ̸ = NP f ∩co-NP f for most functions f: R → ord, although we provide examples where P f = NP f ∩co-NP f and P f ̸ = NP f.

• 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. We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinite-time Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.

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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. 1