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Joining non-low C.E. sets with diagonally non-computable functions

2013
*
Journal of Logic and Computation
*

We show that every

doi:10.1093/logcom/ext039
fatcat:xiq5gdto75a7zn5fghkoyqu5du
*non*-*low**c.e*.*set**joins*all ∆ 0 2*diagonally*noncomputable*functions*to ∅ . ... We give two proofs: a direct argument, and a proof using an analysis of*functions*that are DNC relative to an oracle, extending work by Day and Reimann. ... We extend their result to the wider class of*diagonally**non*-*computable**functions*. Theorem 1.2. If A is a*non*-*low**c.e*.*set*and f is a ∆ 0 2*diagonally**non*-*computable**function*, then ∅ T A ⊕ f . ...##
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Closed choice and a Uniform Low Basis Theorem

2012
*
Annals of Pure and Applied Logic
*

We also prove that all these classes correspond to classes of

doi:10.1016/j.apal.2011.12.020
fatcat:gyfdlmu575ggdcmmypxetxrdl4
*non*-deterministically*computable**functions**with*the respective spaces as advice spaces. ... Finally, we also study the related class of*low**computable**functions*, which contains the class of weakly*computable**functions*as well as the class of*functions**computable**with*finitely many mind changes ... It is known and easy to see that the setA := {f ∈ {0, 1} N : f is two-valued*diagonally**non*-*computable*}is a co-*c.e*. closed*set*. ...##
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Hypersimplicity and semicomputability in the weak truth table degrees

2005
*
Archive for Mathematical Logic
*

Moreover, we consider the

doi:10.1007/s00153-005-0288-9
fatcat:pcmvvqyowndvjis2hnne22q5a4
*sets*that are both hypersimple and semicomputable, characterize them as the (bi-infinite)*c.e*. cuts of*computable*orderings of N of order type ω + ω * and study their wtt degrees ... We construct degrees that are not bounded by hypersimple degrees outside any*non*-trivial upper cone of Turing degrees and show that the hypersimple-free*c.e*. wtt degrees are downwards dense in the*c.e*. ... We are going to construct a*non*-*computable**c.e*. A ≤ wtt C and equivalent to no hypersimple*set*. ...##
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On low for speed oracles
[article]

2017
*
arXiv
*
pre-print

connections

arXiv:1712.09710v1
fatcat:4e2d6otksbhbhcgr6ce2jca64m
*with**computability*theory. ... An oracle A is*low*for speed if relativizing to A has essentially no effect on*computational*complexity, meaning that if a decidable language can be decided in time f(n)*with*access to oracle A, then it ... This paper grew from interactions between complexity theory and classical*computability*theory originating from the 2012 Dagstuhl Seminar "*Computability*, Complexity and Randomness" (Seminar 12012). ...##
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CAPPABLE CEA SETS AND RAMSEY'S THEOREM

2011
*
Proceedings of the 11th Asian Logic Conference
*

This paper introduces the notion of c-cappability and shows that this property cannot be used to obtain such a separation when combined

doi:10.1142/9789814360548_0007
fatcat:cyarjuwsbjaxhegpoac2vkamou
*with*2-CEA-ness. ... We next note that Ding and Qian [4] showed that there is a*non*-zero*c.e*. degree a and a*non*-zero d-*c.e*. degree b whose meet is 0 and whose*join*is 0 . ... Let B 2 be*c.e*. relative to a*c.e*.*set*B 1*with*a*computable*approximation B s 2 as described above. ...##
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Cuppability of Simple and Hypersimple Sets

2007
*
Notre Dame Journal of Formal Logic
*

For

doi:10.1305/ndjfl/1187031408
fatcat:5tfcxuh5u5cqdhpngi7z3xvxae
*sets*fulfilling some type of simplicity property one can now ask whether these*sets*are cuppable*with*respect to a certain type of reducibilities. Several such results are known. ... An incomplete degree is cuppable if it can be*joined*by an incomplete degree to a complete degree. ... For*sets**with*thinner complements this is no longer true as we will see presently. A*set*A is dense immune if the*function*p A enumerating it in order dominates every total*computable**function*. ...##
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On the number of infinite sequences with trivial initial segment complexity

2011
*
Theoretical Computer Science
*

We also show the same for the hierarchy of the

doi:10.1016/j.tcs.2011.09.020
fatcat:rghx53ivwrgf5l3vmeoednhnre
*low*for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the*computable*oracles. ... We show that the problem of finding the number of K -trivial*sets*in the various levels of the hierarchy is ∆ 0 3 . ... In the same paper it was shown that a*set*is complex iff it wtt-*computes*a*diagonally**non*-*computable**function*. Proof. We combine the argument of Theorem 2.14*with**diagonalization*. ...##
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Cupping with random sets
[article]

2012
*
arXiv
*
pre-print

We prove that a

arXiv:1206.1603v1
fatcat:csnbd45sbvgwlpwchq4oo4bmaa
*set*is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a*set*below zero jump is K-trivial if and only if it is not ML-cuppable. ... Any Martin-Löf random*computes*a*diagonally**non*-*computable**function*. ... Hence no Martin-Löf random*set*below ∅ ′ forms a minimal pair*with*any*set*A below ∅ ′ that*computes*a*diagonally**non*-*computable**function*. ...##
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Cupping with random sets

2014
*
Proceedings of the American Mathematical Society
*

We prove that a

doi:10.1090/s0002-9939-2014-11997-6
fatcat:gjvokmdgkbgbhcm2vtx46qxxn4
*set*is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a*set*below zero jump is K-trivial if and only if it is not ML-cuppable. ... Hence no Martin-Löf random*set*below ∅ forms a minimal pair*with*any*set*A below ∅ that*computes*a*diagonally**non*-*computable**function*. ... Any Martin-Löf random*computes*a*diagonally**non*-*computable**function*. ...##
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The search for natural definability in the Turing degrees

2018
*
Computability - The Journal of the Assosiation
*

Theorem 13.3 ([AL4]) Above every

doi:10.3233/com-170068
fatcat:t6qdc3cd6zbadg5ixwoy6jxk7e
*low**c.e*. degree, there is a*low**c.e*. degree which doesn't satisfy*join*. ... In fact, we can suppose not only that B is*non*-*computable*, but also that it is not*c.e*. (in any case, lettingB be the complement of B, B ⊕B is not*c.e*. if B is*non*-*computable*). ...##
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Randomness and Computability: Open Questions

2006
*
Bulletin of Symbolic Logic
*

All

doi:10.2178/bsl/1154698740
fatcat:int7vbtqbnggtfkbbd3wodxo2q
*sets*will be*sets*of natural numbers, unless otherwise stated. These*sets*are identified*with*infinite strings over {0, 1}. Other terms used in the literature are sequence and real. ... It is time for a new paper about open questions in the currently very active area of randomness and*computability*. Ambos-Spies and Kučera presented such a paper in 1999 [1]. ... There is a*non*-*computable**c.e*.*low*for weakly 2-random*set*, and each*low*for weakly 2-random*set*is K-trivial [9] . 7.3. Effective descriptive*set*theory. ...##
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The weakness of being cohesive, thin or free in reverse mathematics

2016
*
Israel Journal of Mathematics
*

In this sense, Ramsey's theorem is not robust

doi:10.1007/s11856-016-1433-3
fatcat:pgo47jxmibgxfaytzaznaj3mia
*with*respect to his number of colors over*computable*reducibility. ... This analysis enables us to answer some questions of Wang about how typical*sets*help*computing*cohesive*sets*. ... The author is funded by the John Templeton Foundation project Structure and Randomness in the Theory of*Computation*(Grant 48003). ...##
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The weakness of being cohesive, thin or free in reverse mathematics
[article]

2016
*
arXiv
*
pre-print

In this sense, Ramsey's theorem is not robust

arXiv:1502.03709v4
fatcat:pq72vylp7zezvhjz5xqhcsivjy
*with*respect to his number of colors over*computable*reducibility. ... This analysis enables us to answer some questions of Wang about how typical*sets*help*computing*cohesive*sets*. ... The author is funded by the John Templeton Foundation ('Structure and Randomness in the Theory of*Computation*' project). ...##
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Bounded Randomness
[chapter]

2012
*
Lecture Notes in Computer Science
*

We introduce some new variations of the notions of being Martin-Löf random where the tests are all clopen

doi:10.1007/978-3-642-27654-5_5
fatcat:zlghgyhwpvdvvd3cu4zxgrzi4i
*sets*. ...*Lowness*Theorem 3. There is a*non*-*computable*Δ 0 3*set*A which is*low*for CBrandomness. Proof (Sketch of proof ). The construction involves building a Δ 0 3 approximation to A. ... Recall that a degree a is array*non*-*computable*2 if for every*function*f ≤ wtt ∅ there is a*function*g ≤ T a such that f (x) < g(x) infinitely often. ...##
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Computing K-Trivial Sets by Incomplete Random Sets
[chapter]

2013
*
Lecture Notes in Computer Science
*

*c.e*.

*sets*are close to being

*computable*. ... The lower and upper cones of noncomputable

*c.e*.

*sets*are definable null

*sets*, and thus if a

*set*is "sufficiently" random, it cannot

*compute*, nor be

*computed*by, a noncomputable

*c.e*.

*set*. ... presented, null G δ class. 1 Any random

*set*

*computes*a

*diagonally*noncomputable

*function*. ...

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