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Infinite time computable model theory
"... Abstract. 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 se ..."
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Abstract. 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. 1.
Space bounds for infinitary computation
"... Infinite Time Turing Machines (or HamkinsKidder machines) have been introduced in [HaLe00] and their computability theory has been investigated in comparison to the usual computability theory in a sequence of papers by Hamkins, Lewis, Welch and Seabold: [HaLe00], [We00a], [We00b], [HaSe01], [HaLe02 ..."
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Infinite Time Turing Machines (or HamkinsKidder machines) have been introduced in [HaLe00] and their computability theory has been investigated in comparison to the usual computability theory in a sequence of papers by Hamkins, Lewis, Welch and Seabold: [HaLe00], [We00a], [We00b], [HaSe01], [HaLe02], [We04], [We05] (cf. also the survey papers [Ha02], [Ha04] and [Ha05]). Infinite Time Turing Machines have the same hardware as ordinary Turing Machines, and almost the same software. However, an Infinite Time Turing Machine can continue its computation if it still hasn’t reached the Halt state after infinitely many steps (for details, see § 1). In [Sc03], Schindler started the investigation of the corresponding complexity theory by defining natural time complexity classes for Infinite Time Turing Machines. Schindler, Welch, Hamkins and Deolalikar have proved with methods of descriptive set theory that the big open questions of standard complexity theory P? = NP and P? = NP ∩ coNP have negative answers for Infinite Time Turing Machines [Sc03,DeHaSc05,HaWe03].
ITTMs with Feedback
, 2009
"... 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 ..."
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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
Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter.
"... We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writabl ..."
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We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writable ordinals. Furthermore, we show that O + is computably isomorphic to h, the lightface halting problem of infinite time Turing machine computability, and that O ++ is computably isomorphic to s, the set of programs that eventually write a real. The last of these results implies, by work of Welch, that O ++ is computably isomorphic to the Σ2 theory of Lζ, and, by work of Burgess, that O ++ is complete with respect to the class of the arithmetically quasiinductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of Σ2(Lζ), a theory in which O ++ plays the role of O, the arithmetically quasiinductive sets play the role of Π1 1, and the eventually writable reals play the role of ∆1 1. 1