Liang Yu

Liang Yu
Nanjing University | NJU · Department of Mathematics

Ph.D
A mathematical logician

About

70
Publications
3,721
Reads
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472
Citations
Citations since 2016
18 Research Items
165 Citations
2016201720182019202020212022051015202530
2016201720182019202020212022051015202530
2016201720182019202020212022051015202530
2016201720182019202020212022051015202530
Additional affiliations
December 2010 - present
Nanjing University
Position
  • Professor (Full)
September 2006 - November 2010
Nanjing University
Position
  • Professor (Associate)
July 2005 - August 2006
National University of Singapore
Position
  • Research Associate
Education
September 1998 - January 2002
Nanjing University
Field of study
  • Mathematics
September 1994 - June 1998
Nanchang University
Field of study
  • Mathematics

Publications

Publications (70)
Article
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We show that the structure ℛ of recursively enumerable degrees is not a Σ₁-elementary substructure of 풟<sub>n</sub>, where 풟<sub>n</sub> (n>1) is the structure of n-r.e. degrees in the Ershov hierarchy.
Article
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One approach to understanding the fine structure of initial seg-ment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤ K Y to mean that (∀n) K(X n) ≤ K(Y n) + O(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence,...
Article
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We give a new proof of Friedman's conjecture that every uncountable Δ11 set of reals has a member of each hyperdegree greater than or equal to the hyperjump.
Data
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We study the strengths of various notions of higher randomness: (i) strong ${\rm{\Pi }}_1^1$ randomness is separated from ${\rm{\Pi }}_1^1$ randomness; (ii) the hyperdegrees of ${\rm{\Pi }}_1^1$ random reals are closed downwards (except for the trivial degree); (iii) the reals z in $NC{R_{{\rm{\Pi }}_1^1}}$ are precisely those satisfying $...
Article
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Assuming ZF, we prove that Turing determinacy (TD) implies the countable choice axiom for sets of reals (CCR).
Preprint
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We prove that, assuming $\mathrm{ZF}$, and restricted to any pointed set, Chaitin's $\Omega_U:x\mapsto \Omega_U^x=\sum_{U^x(\sigma)\downarrow}2^{-|\sigma|}$ is not injective for any universal prefix-free Turing machine $U$, and that $\Omega_U^x$ fails to be degree invariant in a very strong sense, answering several recent questions in descriptive s...
Preprint
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Strongly Turing determinacy, or $\mathrm{sTD}$, says that for any set $A$ of reals, if $\forall x\exists y\geq_T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing determinacy ($\mathrm{TD}$) and $\mathrm{sTD}$: (1). $\mathrm{ZF+TD}$ implies weakly dependent choice ($\mathrm{wDC}$). (2). $\mathrm{...
Preprint
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Assuming $\mathrm{ZF}$, we prove that Turing determinacy ($\mathrm{TD}$) implies countable choice axiom for sets of reals ($\mathrm{CCR}$).
Article
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We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property (N) if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1({\mathcal {O}})$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not...
Preprint
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We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $\Delta^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-L\"of randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness...
Article
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Mauldin [15] proved that there is an analytic set which cannot be represented by B∪X for some Borel set B and a subset X of a Σ02-null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated σ-ideals having approximation property under the assum...
Article
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We prove that the continuous function ${\rm{\hat \Omega }}:2^\omega \to $ that is defined via $X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^...
Preprint
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We prove that the continuous function Ω : 2 ω → R that is defined via X → n 2 −K(X n) for all X ∈ 2 ω is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that 1 0 Ω(X) dX is a left-c.e. wtt-complete real having effective Hausdorff dimension 1 /2. We further investigate the a...
Preprint
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We show that for any x, U h (x) = {y | y ≥ h x} is Borel if and only if x ∈ L. Moreover, the Borel rank of U h (x) when it exists is less than (ω 1) L. We also show using Steel forcing with tagged trees that Borel ranks of these sets range unboundedly below (ω 1) L .
Article
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We prove the following two basis theorems for ${\rm{\Sigma }}_2^1$ -sets of reals: (1) Every nonthin ${\rm{\Sigma }}_2^1$ -set has a perfect ${\rm{\Delta }}_2^1$ -subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$ -subset, and this is equivalent to the statement that there is a nonconstructible real. (2) Every uncountable ${\rm{\Sigma }}_2...
Article
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Let an oracle be called low for prefix-free complexity on a set in case access to the oracle improves the prefix-free complexities of the members of the set at most by an additive constant. Let an oracle be called weakly low for prefix-free complexity on a set in case the oracle is low for prefix-free complexity on an infinite subset of the given s...
Preprint
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We give an answer to an old question by Johnson [9] via a recursion theoretical method. I.e. there is an analytic set which cannot be represented by B ∪ X for some Borel set B and a subset X of a Σ 0 2-null set. We also give a characterization of the Borel generated σ-ideals having approximation property under the assumption that every real is coun...
Article
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We construct a model of ZF + DC containing a Luzin set, a Sierpinski set, as well as a Burstin basis but in which there is no well ordering of the continuum.
Preprint
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We construct a model of $\mathsf{ZF} + \mathsf{DC}$ containing a Luzin set, a Sierpi\'{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.
Article
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In ZF, the existence of a Hamel basis does not yield a well–ordering of ℝ.
Chapter
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We investigate which reals can never be L-random. That is to give a description of the reals which are always belong to some L[λ]-null set for any continuous measure λ. Among other things, we prove that NCRL is an L-cofinal subset of Q3 under ZFC + PD.
Article
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We investigate measure-theoretic aspects of various notions of reducibility by applying analogs of Demuth?s Theorem in the hyperarithmetic and set-theoretic settings.
Article
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The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X -recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a)<p(a)p(a)...
Article
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We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT).
Article
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Assuming , we prove that holds if and only if there exists a cofinal maximal chain of order type in the Turing degrees. However, it is consistent that +``the reals are not well ordered''+``there exists a cofinal chain in the Turing degrees of order type ''. - See more at: http://www.ams.org/journals/proc/2014-142-04/S0002-9939-2014-11868-5/#sthash....
Conference Paper
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We explore the computational strength of the hyperimmune-free Turing degrees. In particular we investigate how the property of being dominated by recursive functions interacts with classical computability notions such as the jump operator, relativization and effectively closed sets.
Article
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We introduce two methods for characterizing strong randomness notions via Martin-Löf randomness. We apply these methods to investigate Schnorr randomness relative to 0̸′.
Article
We study oscillation in the prefix-free complexity of initial segments of 1-random reals. For upward oscillations, we prove that ∑n∈ω2−g(n) diverges iff (∃∞n)K(X↾n)>n+g(n) for every 1-random X∈ω2. For downward oscillations, we characterize the functions g such that (∃∞n)K(X↾n)<n+g(n) for almost every X∈ω2. The proof of this result uses an improveme...
Article
Computably Lipschitz reducibility (noted as ≤cl for short), was suggested as a measure of relative randomness. We say α≤clβ if α is Turing reducible to β with oracle use on x bounded by x+c. In this paper, we prove that for any non-computable Δ20 real, there exists a c.e. real so that no c.e. real can cl-compute both of them. So every non-computabl...
Article
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We study the descriptive set theoretical complexity of various randomness notions.
Article
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A real xx is Δ11-Kurtz random (Π11-Kurtz random) if it is in no closed null Δ11 set (Π11 set). We show that there is a cone of Π11-Kurtz random hyperdegrees. We characterize lowness for Δ11-Kurtz randomness as being Δ11-dominated and Δ11-semi-traceable.
Article
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We show that Martin's conjecture on Π 1 1 -functions uniformly ≤ T -order preserving on a cone implies Π 1 1 Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant Π 1 2n+1 -functions is equivalent over ZFC to Σ 1 2n+2 -Axiom of Determinacy. As a corollary, the consistency of t...
Article
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We prove that every Turing degree a bounding some non-GL₂ degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.
Article
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We prove that every Turing degree a bounding some non-GL(2) degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.
Article
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We introduce a 11-uniformization principle and establish its equiva- lence with the set-theoretic hypothesis (!1)L = !1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of 11 maximal chains and thin maximal antichains in the Turing degrees. We also use the 11-uniformization principle...
Article
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An n-r.e. set can be defined as the symmetric difference of n recursively enumerable sets, The classes of these sets form a natural hierarchy which became a well-studied topic in recursion theory. In a series of ground-breaking papers, Ershov generalized this hierarchy to transfinite levels based on Kleene's notations of ordinals and this work lead...
Article
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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. 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...
Article
Conference on Computability, Complexity and Randomness - Volume 14 Issue 4 - Jinhe Chen, Decheng Ding, Liang Yu
Article
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We study randomness notions given by higher recursion theory, establishing the relationships Π11-randomness ⊂ Π11-Martin-Löf randomness ⊂ Δ11-randomness = Δ11-Martin-Löf randomness. We characterize the set of reals that are low for Δ11 randomness as precisely those that are Δ11-traceable. We prove that there is a perfect set of such reals.
Article
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We study the dierences among finite levels of the Ershov hierarchies. We also give a brief survey on the current state of this area. Some questions are raised.
Article
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Assuming ZFC, we prove that CH holds if and only if there exists a cofinal maximal chain of order type !1 in the Turing degrees. 1. Introduction and Notations Introduction. A chain in the Turing degrees is a set of degrees in which any two distinct elements are Turing comparable. A maximal chain is a chain which cannot be properly extended. A chain...
Article
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We study the problem of existence of maximal chains in the Turing degrees. We show that:1. ZF + DC + “There exists no maximal chain in the Turing degrees” is equiconsistent with ZFC + “There exists an inaccessible cardinal”2. For all a ∈ 2 ω , (ω 1) L[a] = ω 1 if and only if there exists a [a] maximal chain in the Turing degrees. As a corollary, ZF...
Conference Paper
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We study existence problems of maximal antichains in the Turing de- grees. In particular, we give a characterization of the existence of a thin 11 maximal antichains in the Turing degrees in terms of (relatively) constructible reals. A corol- lary of our main result gives a negative solution to a question of Jockusch under the assumption that every...
Article
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We prove that van Lambalgen's Theorem fails for both Schnorr randomness and computable randomness. To characterize randomness, various definitions of randomness for individual elements of Cantor space have been introduced. The most popular (and maybe the most important) definitions of randomness are Martin-Lof randomness, Schnorr randomness and com...
Article
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We prove that for any locally countable partial order ℙ = (2ε, ≤ p , there exists a nonmeasurable antichain in ℙ. Some applications of the result are also presented.
Article
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We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.
Article
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We answer a question of Jockusch by constructing a hyperimmune- free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions...
Conference Paper
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We prove that a set is low for weakly 1-generic i it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Fur- thermore, we show that every set which is low for weakly 1-generic is also low for Kurtz-random.
Conference Paper
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We study the differences among elementary theories of finite levels of Ershov hierarchies. We also give a brief survey on the current state of this area. Some questions are raised.
Article
We study lowness for genericity. We show that there exists no Turing degree which is low for 1-genericity and all of computably traceable degrees are low for weak 1-genericity.
Article
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Let [NB]<sub>1</sub> denote the ideal generated by nonbounding c.e. degrees and NCup the ideal of noncuppable c.e. degrees. We show that both [NB]<sub>1</sub> ∩ NCup and the ideal generated by nonbounding and noncuppable degrees are new, in the sense that they are different from M, [NB]<sub>1</sub> and NCup—the only three known definable ideals so...
Article
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We prove that there is no sw-complete c.e. real, negatively answering a question in [6].
Article
We investigate the initial segment complexity of random reals. Let K(σ) denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K(α↾n) and K(β↾n). It is well-known that a real α is 1-random iff there is a constant c such that for all n, K(α↾n)⩾n−c. We ask the question, wha...
Article
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We prove that there are 2@0 many H-degrees in the random reals.
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We prove that there is no a maximal low d.c.e degree.
Article
We prove that there are non-recursive r.e. sets A and C with A <T C such that for every set \( F \leqslant _{T} A,{\kern 1pt} {\kern 1pt} C \cap F \equiv _{W} \emptyset \).
Article
(i) Call a c.e. degreeb anti-cupping relative tox, if there is a c.e.a<b such that for any c.e.w, w≱x impliesa∪w≱∪x. (ii) Call a c.e. degreeb everywhere anti-cupping (e.a.c.), if it is anti-cupping relative tox for each c.e. degreex. By a tree method, we prove that every high c.e. degree has e.a.c. property by extending Harrington’s anti-cupping...
Article
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We prove that lowness for weak genericity is equivalent to semi-computable traceability which is strictly between hyperimmune-freeness and computable traceability. We also show that semi-computable traceability im-plies lowness for weak randomness. These results refute a conjecture raised by several people.
Article
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We show that the structure R of recursively enumerable degrees is not a 1-elementary substructure of Dn, where Dn (n > 1) is the structure of n-r.e. degrees in Ershov hierarchy.
Article
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We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Lof randomness.
Article
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A computable presentation of the linearly ordered set (!; ), where ! is the set of natural numbers and is the natural order on !, is any linearly ordered set L = (!; L) isomorphic to (!; ) such that L is a computable relation. Let X be subset of ! and XL be the image of X in the linear order L under the isomorphism between (!; ) and L. The degree s...

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