Chaitin Ω Numbers and Halting Problems

Tadaki, Kohtaro

doi:10.1007/978-3-642-03073-4_46

Cited by 7 publications

(14 citation statements)

References 15 publications

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“…We conclude this section with a brief discussion of some results from Tadaki [Tad09] which came to our attention only recently. Tadaki's results are incomparable with those presented here, but relate to the results from this paper concerning reductions between Ω and c.e.…”

Section: Omega Numbers and Completenessmentioning

confidence: 93%

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Barmpalias

Fang

Lewis-Pye

2016

Journal of Computer and System Sciences

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Abstract. Chaitin's number Ω is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real α, there exists a Turing functional via which Ω computes α, and such that the number of bits of Ω that are needed for the computation of the first n bits of α (i.e. the use on argument n) is bounded above by a computable function h(n) = n + o (n).We characterise the asymptotic upper bounds on the use of Chaitin's Ω in oracle computations of halting probabilities (i.e. c.e. reals). We show that the following two conditions are equivalent for any computable function h such that h(n) − n is non-decreasing: (1) h(n) − n is an information content measure, i.e. the series n 2 n−h(n) converges, (2) for every c.e. real α there exists a Turing functional via which Ω computes α with use bounded by h. We also give a similar characterisation with respect to computations of c.e. sets from Ω, by showing that the following are equivalent for any computable non-decreasing function g: (1) g is an information-content measure, (2) for every c.e. set A, Ω computes A with use bounded by g. Further results and some connections with Solovay functions (studied by a number of authors [Sol75, BD09, HKM09, BMN11]) are given.

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Section: Omega Numbers and Completenessmentioning

confidence: 93%

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Barmpalias

Fang

Lewis-Pye

2016

Journal of Computer and System Sciences

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“…Tadaki [96] improved on the latter result by showing the following characterization. 20 Given optimal machines W, V and a computable function f , we have i 2 − f (i) < ∞ if and only if D n (W) uniformly computes the first n − f (n) − c bits of Ω V for some constant c.…”

Section: Omega and Halting Problemsmentioning

confidence: 89%

“…In the same paper it was shown that every halting set with respect to a Kolmogorov numbering computes Omega with use O (2 n ). The reader may compare these results with Tadaki [96] who considered halting sets W of universal prefix-free machines, instead of the more compact halting sets of universal plain Turing machines. It was shown that in the prefix-free case, about 2 n+2 log n bits of W are needed for the computation of Ω ↾ n , and that 2 n+log n do not always suffice.…”

Section: Analogues Of Omega In the Computably Enumerable Setsmentioning

confidence: 99%

Aspects of Chaitin's Omega

Barmpalias

2017

Preprint

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The halting probability of a Turing machine, also known as Chaitin's Omega, is an algorithmically random number with many interesting properties. Since Chaitin's seminal work, many popular expositions have appeared, mainly focusing on the metamathematical or philosophical significance of Omega (or debating against it). At the same time, a rich mathematical theory exploring the properties of Chaitin's Omega has been brewing in various technical papers, which quietly reveals the significance of this number to many aspects of contemporary algorithmic information theory. The purpose of this survey is to expose these developments and tell a story about Omega, which outlines its multifaceted mathematical properties and roles in algorithmic randomness.

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“…Asymptotic conditions on the redundancy g in computations from random oracles such as the ones in Theorem 1.4, have been used with respect to Chaitin's Ω in Tadaki [Tad09] and Barmpalias, Fang and Lewis-Pye [BFLP16]. However the latter work only refers to computations of computably enumerable sets and reals and does not have essential connections with the present work, except perhaps for some apparent analogy of the statements proved.…”

Section: Further Related Work In the Literaturementioning

confidence: 99%

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

Barmpalias

Lewis-Pye

Teutsch

2016

Information and Computation

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Abstract. The Kučera-Gács theorem [Kuč85, Gác86] is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n + h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate coding procedure which achieves redundancy √ n log n. A similar bound is implicit in the later proof by Merkle and Mihailović [MM04]. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that n 2 −g(n) = ∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies n 2 −g(n) < ∞. Moreover, the class of such reals is comeager and includes a ∆ 0 2 real as well as all weakly 2-generic reals. On the other hand, it has been recently shown in [BLP16] that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that n 2 −g(n) < ∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.

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Chaitin Ω Numbers and Halting Problems

Cited by 7 publications

References 15 publications

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega

Aspects of Chaitin's Omega

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

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