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

L. Bienvenu, N. Greenberg, A. Kucera, J. S. Miller, A. Nies, D. Turetsky
2013 Journal of Logic and Computation  
We show that every 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 .  ... 
doi:10.1093/logcom/ext039 fatcat:xiq5gdto75a7zn5fghkoyqu5du

Closed choice and a Uniform Low Basis Theorem

Vasco Brattka, Matthew de Brecht, Arno Pauly
2012 Annals of Pure and Applied Logic  
We also prove that all these classes correspond to classes of 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.  ... 
doi:10.1016/j.apal.2011.12.020 fatcat:gyfdlmu575ggdcmmypxetxrdl4

Hypersimplicity and semicomputability in the weak truth table degrees

George Barmpalias
2005 Archive for Mathematical Logic  
Moreover, we consider the 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.  ... 
doi:10.1007/s00153-005-0288-9 fatcat:pcmvvqyowndvjis2hnne22q5a4

On low for speed oracles [article]

Laurent Bienvenu, Rod Downey
2017 arXiv   pre-print
connections 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).  ... 
arXiv:1712.09710v1 fatcat:4e2d6otksbhbhcgr6ce2jca64m

CAPPABLE CEA SETS AND RAMSEY'S THEOREM

Asher M. Kach, Manuel Lerman, Reed Solomon
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 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.  ... 
doi:10.1142/9789814360548_0007 fatcat:cyarjuwsbjaxhegpoac2vkamou

Cuppability of Simple and Hypersimple Sets

Martin Kummer, Marcus Schaefer
2007 Notre Dame Journal of Formal Logic  
For 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.  ... 
doi:10.1305/ndjfl/1187031408 fatcat:5tfcxuh5u5cqdhpngi7z3xvxae

On the number of infinite sequences with trivial initial segment complexity

George Barmpalias, T.F. Sterkenburg
2011 Theoretical Computer Science  
We also show the same for the hierarchy of the 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.  ... 
doi:10.1016/j.tcs.2011.09.020 fatcat:rghx53ivwrgf5l3vmeoednhnre

Cupping with random sets [article]

Adam R. Day, Joseph S. Miller
2012 arXiv   pre-print
We prove that a 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.  ... 
arXiv:1206.1603v1 fatcat:csnbd45sbvgwlpwchq4oo4bmaa

Cupping with random sets

Adam R. Day, Joseph S. Miller
2014 Proceedings of the American Mathematical Society  
We prove that a 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.  ... 
doi:10.1090/s0002-9939-2014-11997-6 fatcat:gjvokmdgkbgbhcm2vtx46qxxn4

The search for natural definability in the Turing degrees

Andrew E.M. Lewis-Pye, Benedikt Löwe
2018 Computability - The Journal of the Assosiation  
Theorem 13.3 ([AL4]) Above every 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).  ... 
doi:10.3233/com-170068 fatcat:t6qdc3cd6zbadg5ixwoy6jxk7e

Randomness and Computability: Open Questions

Joseph S. Miller, André Nies
2006 Bulletin of Symbolic Logic  
All 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.  ... 
doi:10.2178/bsl/1154698740 fatcat:int7vbtqbnggtfkbbd3wodxo2q

The weakness of being cohesive, thin or free in reverse mathematics

Ludovic Patey
2016 Israel Journal of Mathematics  
In this sense, Ramsey's theorem is not robust 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).  ... 
doi:10.1007/s11856-016-1433-3 fatcat:pgo47jxmibgxfaytzaznaj3mia

The weakness of being cohesive, thin or free in reverse mathematics [article]

Ludovic Patey
2016 arXiv   pre-print
In this sense, Ramsey's theorem is not robust 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).  ... 
arXiv:1502.03709v4 fatcat:pq72vylp7zezvhjz5xqhcsivjy

Bounded Randomness [chapter]

Paul Brodhead, Rod Downey, Keng Meng Ng
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 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.  ... 
doi:10.1007/978-3-642-27654-5_5 fatcat:zlghgyhwpvdvvd3cu4zxgrzi4i

Computing K-Trivial Sets by Incomplete Random Sets [chapter]

Noam Greenberg
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.  ... 
doi:10.1007/978-3-642-39053-1_26 fatcat:ldxmn62dznbddlkmk6mv5ihfem
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