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Randomness and Computability: Open Questions

2006
*
Bulletin of Symbolic Logic
*

When we think

doi:10.2178/bsl/1154698740
fatcat:int7vbtqbnggtfkbbd3wodxo2q
*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*. ...##
###
Lowness for Weakly 1-generic and Kurtz-Random
[chapter]

2006
*
Lecture Notes in Computer Science
*

Furthermore, it

doi:10.1007/11750321_72
fatcat:jwr7lpb6abepnhvisqi6oj5ghe
*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. ...##
###
Relativizations of randomness and genericity notions

2011
*
Bulletin of the London Mathematical Society
*

It

doi:10.1112/blms/bdr007
fatcat:cbslvdcenbc7jdpj6ctill6tou
*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*. ...##
###
Relativized Schnorr tests with universal behavior

2010
*
Archive for Mathematical Logic
*

We prove

doi:10.1007/s00153-010-0187-6
fatcat:ergn4dyl5falrb4dllavr5dkji
*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. ...##
###
Degrees of Weakly Computable Reals
[chapter]

2006
*
Lecture Notes in Computer Science
*

-There are nonrecursive

doi:10.1007/11780342_43
fatcat:roqnxjutfvfhtn7k7ehqto2oyq
*sets**A*such*that*Ω*is**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*. ...##
###
An Analogy between Cardinal Characteristics and Highness Properties of Oracles

2015
*
Proceedings of the 13th Asian Logic Conference
*

We develop an analogy between cardinal characteristics from

doi:10.1142/9789814678001_0001
fatcat:q66pryjhjrgz7grfsxu5ammfze
*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*. ...##
###
An analogy between cardinal characteristics and highness properties of oracles
[article]

2014
*
arXiv
*
pre-print

We present an analogy between cardinal characteristics from

arXiv:1404.2839v2
fatcat:hkg2eyqdvfhetkqfltfxiijbsu
*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*...##
###
Demuth randomness and computational complexity

2011
*
Annals of Pure and Applied Logic
*

We show

doi:10.1016/j.apal.2011.01.004
fatcat:4utwog6c5zcwvgc3qoefjouo54
*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. ...##
###
Two More Characterizations of K-Triviality

2018
*
Notre Dame Journal of Formal Logic
*

Nies [11]

doi:10.1215/00294527-2017-0021
fatcat:2xineptjwbbgliowzbau7serca
*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). ...##
###
Difference randomness

2011
*
Proceedings of the American Mathematical Society
*

the Demuth

doi:10.1090/s0002-9939-2010-10513-0
fatcat:ttiafmn525eddnhmiy7sjjifmy
*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*. ...##
###
Genericity and UD-random reals

2015
*
Journal of Logic and Analysis
*

We prove

doi:10.4115/jla.2015.7.4
fatcat:h7gtmm73efe6foc5gedifdxx6a
*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 ...##
###
Committee-based Selection of Weakly Labeled Instances for Learning Relation Extraction

2013
*
Research in Computing Science
*

This approach

doi:10.13053/rcs-70-1-14
fatcat:kn5ykwbntrebhi22fg7kpely7a
*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. ...##
###
Weakly Supervised Classification of Objects in Images Using Soft Random Forests
[chapter]

2010
*
Lecture Notes in Computer Science
*

This paper deals with

doi:10.1007/978-3-642-15561-1_14
fatcat:d7tsp3opqjer3hjwngxjiiczmi
*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. ...##
###
HIGHER RANDOMNESS AND GENERICITY

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 . ...

##
###
Randomness notions and partial relativization

2012
*
Israel Journal of Mathematics
*

We study the computational complexity of an oracle

doi:10.1007/s11856-012-0012-5
fatcat:tn35aaki65e25ayorjb6gkskoq
*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. ...
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