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Randomness and Computability: Open Questions

Joseph S. Miller, André Nies
2006 Bulletin of Symbolic Logic  
When we think a question is a major one, and therefore likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers  ...  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].  ...  They also show that each low for weakly 1-generic is low for weakly random.  ... 
doi:10.2178/bsl/1154698740 fatcat:int7vbtqbnggtfkbbd3wodxo2q

Lowness for Weakly 1-generic and Kurtz-Random [chapter]

Frank Stephan, Liang Yu
2006 Lecture Notes in Computer Science  
Furthermore, it is shown that every set which is low for weakly 1-generic is also low for Kurtz-random.  ...  It is shown that a set is low for weakly 1-generic iff it has neither dnr nor hyperimmune Turing degree.  ...  Further thanks go to Rod Downey for inviting the first author to present the results of this work in a talk of the special session on randomness of the conference TAMC 2006.  ... 
doi:10.1007/11750321_72 fatcat:jwr7lpb6abepnhvisqi6oj5ghe

Relativizations of randomness and genericity notions

Johanna N. Y. Franklin, Frank Stephan, Liang Yu
2011 Bulletin of the London Mathematical Society  
It is shown that A is a basis for Schnorr randomness if and only if A is a basis for weak 1-genericity if and only if the halting problem K is not Turing reducible to A.  ...  A set A is a basis for Schnorr randomness if and only if it is Turing reducible to a set R which is Schnorr random relative to A. One can define a basis for weak 1-genericity similarly.  ...  The authors would like to thank Joe Miller for his correspondence and the information on his result concerning highness for Martin-Löf randomness versus strong randomness.  ... 
doi:10.1112/blms/bdr007 fatcat:cbslvdcenbc7jdpj6ctill6tou

Relativized Schnorr tests with universal behavior

Nicholas Rupprecht
2010 Archive for Mathematical Logic  
We prove that the sets with this property are exactly those with high Turing degree.  ...  Our method is closely related to the proof of Terwijn and Zambella's characterization of the oracles which are low for Schnorr tests.  ...  A real A is low for weak 1-genericity if every B which is weakly 1-generic is weakly 1-generic relative to A. Definition 5. A real B is weakly 1-generic (relative to Remark.  ... 
doi:10.1007/s00153-010-0187-6 fatcat:ergn4dyl5falrb4dllavr5dkji

Degrees of Weakly Computable Reals [chapter]

Keng Meng Ng, Frank Stephan, Guohua Wu
2006 Lecture Notes in Computer Science  
-There are nonrecursive sets A such thatis random relative to A. These sets are called low for Ω.  ...  Then there is a function f ≤ T K such that: (1) every f -limit-generic set A is 1-generic, (2) for all i, if α i ≤ T A, then α i is recursive, (3) for all i, if α i ≥ T A, then α i is complete. (4) Furthermore  ...  There is a 1-generic degree below 0 such that every nonzero degree below it contains no weakly computable reals. 2.  ... 
doi:10.1007/11780342_43 fatcat:roqnxjutfvfhtn7k7ehqto2oyq

An Analogy between Cardinal Characteristics and Highness Properties of Oracles

Jörg Brendle, Andrew Brooke-Taylor, Keng Meng Ng, André Nies
2015 Proceedings of the 13th Asian Logic Conference  
We develop an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong.  ...  (9) Is there a degree which is not low for Schnorr tests but which is low for weak 1-genericity and not weakly Schnorr engulfing?  ...  Note that Rupprecht or Thm. 19]. (3) There is a set A which is not weakly Schnorr engulfing and computes a Schnorr random .  ... 
doi:10.1142/9789814678001_0001 fatcat:q66pryjhjrgz7grfsxu5ammfze

An analogy between cardinal characteristics and highness properties of oracles [article]

Jörg Brendle, Andrew Brooke-Taylor, Keng Meng Ng, André Nies
2014 arXiv   pre-print
We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong.  ...  After a comprehensive survey of the analogy for characteristics from Cichon's diagram, we extend it to Kurtz randomness and the analogue of the Specker-Eda number.  ...  (R consists of weakly meager engulfing, not low for weak 1 genericity, and not low for Schnorr tests.) (9) Is there a set which is not low for Schnorr tests, is low for weak 1-genericity and not weakly  ... 
arXiv:1404.2839v2 fatcat:hkg2eyqdvfhetkqfltfxiijbsu

Demuth randomness and computational complexity

Antonín Kučera, André Nies
2011 Annals of Pure and Applied Logic  
We show that a weakly Demuth random set can be high and ∆ 0 2 , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable.  ...  We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function.  ...  The first author is partially supported by the Marsden Fund of New Zealand, grant no. 08-UOA-187.  ... 
doi:10.1016/j.apal.2011.01.004 fatcat:4utwog6c5zcwvgc3qoefjouo54

Two More Characterizations of K-Triviality

Noam Greenberg, Joseph S. Miller, Benoit Monin, Daniel Turetsky
2018 Notre Dame Journal of Formal Logic  
Nies [11] generalized (c) to LR-reducibility: we write A ≤ LR B to mean that every B-random set is A-random. In particular, A ≤ LR ∅ means that A is low for randomness (hence K-trivial).  ...  We prove that if A LR B, then for every set X there is a B-random set Y such that X is computable from Y ⊕ A.  ...  Assume that A LR B. Then for any set X, there is a B-random set Y such that X ≤ T Y ⊕ A (in fact, we make Y weakly 2-random relative to B).  ... 
doi:10.1215/00294527-2017-0021 fatcat:2xineptjwbbgliowzbau7serca

Difference randomness

Johanna N. Y. Franklin, Keng Meng Ng
2011 Proceedings of the American Mathematical Society  
the Demuth random and weakly 2-random reals.  ...  In one case, we call the resulting notion difference randomness and show that it results in a class of random reals that is a strict subclass of the Martin-Löf random reals and a proper superclass of both  ...  Suppose A is an r.e. set. Then A is low for difference randomness if and only if A is not weakly Martin-Löf cuppable. Proof. Suppose that A is an r.e. set that is low for difference randomness.  ... 
doi:10.1090/s0002-9939-2010-10513-0 fatcat:ttiafmn525eddnhmiy7sjjifmy

Genericity and UD-random reals

Wesley Calvert, Johanna Franklin
2015 Journal of Logic and Analysis  
We prove that there exists a weakly 1-random real that is neither UD-random nor weakly 1-generic. We also show that no 2-generic real can Turing compute a UD-random real.  ...  Avigad introduced the notion of UD-randomness based in Weyl's 1916 definition of uniform distribution modulo one.  ...  Proof To construct such an X ∈ 2 ω , we must build not only X itself but also a dense Σ 0 1 set of strings S that X avoids (witnessing that X is not weakly 1-generic) and an infinite recursive sequence  ... 
doi:10.4115/jla.2015.7.4 fatcat:h7gtmm73efe6foc5gedifdxx6a

Committee-based Selection of Weakly Labeled Instances for Learning Relation Extraction

Tamara Bobić, Roman Klinger
2013 Research in Computing Science  
This approach is similar to the active learning paradigm, with a difference that unlabeled instances are weakly annotated, rather than by human experts.  ...  Different strategies using low or high confidence are compared to random selection.  ...  Training on HPRD50 does not provide a clear difference between the selection strategies; low leads to worst results, random and high with some noise to the best.  ... 
doi:10.13053/rcs-70-1-14 fatcat:kn5ykwbntrebhi22fg7kpely7a

Weakly Supervised Classification of Objects in Images Using Soft Random Forests [chapter]

Riwal Lefort, Ronan Fablet, Jean-Marc Boucher
2010 Lecture Notes in Computer Science  
This paper deals with weakly supervised learning that generalizes the supervised and semi-supervised learning.  ...  In weakly supervised learning training data are given as the priors of each class for each sample. We first propose a weakly supervised strategy for learning soft decision trees.  ...  Interestingly the later is equivalent to the C4.5 random forest [15] if a supervised dataset is available such that recognition performances for low uncertainty priors can be granted.  ... 
doi:10.1007/978-3-642-15561-1_14 fatcat:d7tsp3opqjer3hjwngxjiiczmi

HIGHER RANDOMNESS AND GENERICITY

NOAM GREENBERG, BENOIT MONIN
2017 Forum of Mathematics, Sigma  
2-randomness.  ...  We discuss lowness for $\unicode[STIX]{x1D6F1}_{1}^{1}$ -randomness, cupping with $\unicode[STIX]{x1D6F1}_{1}^{1}$ -random sequences, and an analogue of the Hirschfeldt–Miller characterization of weak  ...  We want to show that K is the set of sequences which are not higher weakly 2 random. In one direction, suppose that X is not higher weakly 2 random. Find some e such that λpG e q " 0 and X P G e .  ... 
doi:10.1017/fms.2017.27 fatcat:tzbbv3kjlbecjjkgqviohr5rqa

Randomness notions and partial relativization

George Barmpalias, Joseph S. Miller, André Nies
2012 Israel Journal of Mathematics  
We study the computational complexity of an oracle set using a number of notions of randomness that lie between Martin-Löf randomness and 2-randomness in terms of strength.  ...  We characterize the oracles A such that ML[A] ⊆ C, where C is such a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A, using a new meta-concept called partial relativization  ...  Note that Z is weakly random iff it is not a member of any null Π 0 1 class; Z is weakly 2-random if it is not a member of any null Π 0 2 class.  ... 
doi:10.1007/s11856-012-0012-5 fatcat:tn35aaki65e25ayorjb6gkskoq
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