Randomness and degree theory for infinite time register machines1

Carl, Merlin

doi:10.3233/com-160055

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…We prove a variant of the previous theorem for recognizable sets. 1 Thus we answer several questions posed in [Car15, Section 5] and [Carb,Section 6]. The next result characterizes ITTM-randomness by the values of an ordinal Σ that is associated to ITTM-computations, the supremum of the ordinals coded by accidentally writable reals, i.e.…”

mentioning

confidence: 69%

“…Moreover, they do not use any specific properties of Lebesgue measure and therefore hold for arbitrary measures with the property that the function that maps s ∈ 2 < to (N s ) is computable. Finally, most results in this section hold for genericity instead of randomness and for some other machine models, for instance ITRM-genericity [5].…”

Section: Proof It Was Shown In the Proof Of Lemma 43 That The Setmentioning

confidence: 98%

“…We prove a variant of the previous theorem for recognizable sets. 1 Thus we answer several questions posed in [Car15, Section 5] and [Carb,Section 6].…”

Section: Introductionmentioning

confidence: 96%

“…The strength of these machines is strictly above Π 1 1 and therefore, this motivates the consideration of notions of randomness based on ITTMs. This project was started in [8] and continued in [5,6].…”

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confidence: 99%

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Randomness via Infinite Computation and Effective Descriptive Set Theory

Carl¹,

Schlicht²

2018

J. symb. log.

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We study randomness beyond Π 1 1 -randomness and its Martin-Löf type variant, introduced in [HN07] and further studied in [BGM].The class given by the infinite time Turing machines (ITTMs), introduced by Hamkins and Kidder, is strictly between Π 1 1 and Σ 1 2 . We prove that the natural randomness notions associated to this class have several desirable properties resembling those of the classical random notions such as Martin-Löf randomness, and randomness notions defined via effective descriptive set theory such as Π 1 1 -randomness. For instance, mutual randoms do not share information and can be characterized as in van Lambalgen's theorem. We also obtain some differences to the hyperarithmetic setting. Already at the level of Σ 1 2 , some properties of randomness notions are independent [CS17]. Towards the results about randomness, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool, we prove facts of independent interest about random forcing over admissible sets and increasing unions of admissible sets. These results are also useful for more efficient proofs of some classical results about hyperarithmetic sets.

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confidence: 69%

Section: Proof It Was Shown In the Proof Of Lemma 43 That The Setmentioning

confidence: 98%

“…We prove a variant of the previous theorem for recognizable sets. 1 Thus we answer several questions posed in [Car15, Section 5] and [Carb,Section 6].…”

Section: Introductionmentioning

confidence: 96%

mentioning

confidence: 99%

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Randomness via Infinite Computation and Effective Descriptive Set Theory

Carl¹,

Schlicht²

2018

J. symb. log.

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Some Observations on Infinitary Complexity

Carl

2018

Sailing Routes in the World of Computation

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Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was done in [CLR], we prove the following results:1. An analogue of Ladner's theorem for OTMs holds: That is, there are languages L which are NP ∞ , but neither P ∞ nor NP ∞complete.This answers an open question of [CLR].2. The speedup theorem for Turing machines, which allows us to bring down the computation time and space usage of a Turing machine program down by an aribtrary positive factor under relatively mild side conditions by expanding the working alphabet does not hold for OTMs.3. We show that, for α < β such that α is the halting time of some OTM-program, there are decision problems that are OTMdecidable in time bounded by |w| β · γ for some γ ∈ On, but not in time bounded by |w| α · γ for any γ ∈ On.

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Infinite time recognizability from generic oracles and the recognizable jump operator1

Carl

2017

COM

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By a theorem of Sacks, if a real x is recursive relative to all elements of a set of positive Lebesgue measure, x is recursive. This statement, and the analogous statement for non-meagerness instead of positive Lebesgue measure, have been shown to carry over to many models of transfinite computations in [11]. Here, we start exploring another analogue concerning recognizability rather than computability. For a notion of relativized recognizability (introduced in [6] for ITRMs and generalized here to various other machine types), we show that, for Infinite Time Turing Machines (ITTMs), if a real x is recognizable relative to all elements of a non-meager Borel set Y , then x is recognizable. We also show that a relativized version of this statement holds for Infinite Time Register Machines (ITRMs). This extends our work in [8] where we obtained the (unrelativized) result for ITRMs. We then introduce a jump operator for recognizability, examine its set-theoretical content and show that the recognizable jumps for ITRMs and ITTMs are primitive-recursively equivalent, even though these two models are otherwise of vastly different strength. Finally, we introduce degrees of recognizability by considering the transitive closure of relativized recognizability and connect it with the recognizable jump operator to obtain a solution to Post's problem for degrees of recognizability.

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Randomness and degree theory for infinite time register machines1

Cited by 3 publications

References 22 publications

Randomness via Infinite Computation and Effective Descriptive Set Theory

Randomness via Infinite Computation and Effective Descriptive Set Theory

Some Observations on Infinitary Complexity

Infinite time recognizability from generic oracles and the recognizable jump operator1

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