Computably Enumerable Equivalence Relations

The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\E$, such that the question of membership in this orbit is $Σ^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\E$ is $\wock +1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\E$; for all finite $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof). A few small corrections made in this version

Realizing Computably Enumerable Degrees in Separating Classes

We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.

A note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real $α$ there exists a left-c.e. real $β$ such that $α\neq β+γ$ for all left-c.e. reals and all right-c.e. reals $γ$. The proof is non-uniform, the dichotomy being whether the given real $α$ is Martin-Loef random or not. It follows that given any universal machine $U$, there is another universal machine $V$ such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.

Universal computably enumerable sets and initial segment prefix-free complexity

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment complexity, there exists a Turing complete computably enumerable set $B$ with complexity strictly less than the complexity of $A$. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete. Moreover we give a generalization of this result for any finite collection of computably enumerable sets $A_i, i<k$…

Relatively computably enumerable reals

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.

Asymptotic Density and Computably Enumerable Sets

We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural characterization of non-low c.e. degrees. In contrast, we prove that every nonzero c.e. degree contains a set which is generically computable but not coarsely computable. There is a very close connection between the computational complexity of a set and the computational complexity of its density as a real number where we measure complexity of real numbers as th…

Kolmogorov complexity and computably enumerable sets

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity …

Computably Enumerable Sets that are Automorphic to Low Sets

We work with the structure consisting of all computably enumerable (c.e.) sets ordered by set inclusion. The question we will partially address is which c.e.\ sets are autormorphic to low (or low$_2$ sets. Using work of Miller, we can see that every set with semilow complement is $Δ^0_3$ automorphic to a low set. While it remains open whether every set with semilow complement is effectively automorphic to a low set, we show that there are sets without semilow complement that are effectively automorphic to low sets. We also consider other lowness notions such as having a semilow$_{1.5}$ complem…

Low$_2$ 可计算可枚举集具有超超简单超集

Low$_2$ computably enumerable sets have hyperhypersimple supersets

本文证明 low$_2$ 可计算可枚举集具有无原子的超超简单超集，并进一步推广到任意 $\Sigma_3$-布尔代数作为超集格，解决了该长期猜想的一个主要测试案例。

Turing Degrees and Automorphism Groups of Substructure Lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. In this paper, we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree $\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector spaces, o…

Every Computably Enumerable Random Real Is Provably Computably Enumerable Random

We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that "a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine" can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost …

Extensions Theorems, Orbits, and Automorphisms of the Computably Enumerable Sets

We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal{E}$. Using this extension theorem and other work we then show if $A$ and $\hat{A}$ are automorphic via $Ψ$ then they are automorphic via $Λ$ where $Λ\restriction Ł^*(A) = Ψ$ and $Λ\restriction \E^*(A)$ is $Δ^0_3$. We give an algebraic description of when an arbitrary set $\Ahat$ is in the orbit of a \ce set $A$. We construct the first example of a definable orbit which is not a $Δ^0_3$ orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show …

给定度下的切换范畴性行为

Switching Categoricity Behavior with a Given Degree

针对给定不可计算可枚举度，构造结构实现计算范畴性与相对范畴性的切换，否定相关猜想。

稳定可计算形式系统的不完全性

Incompleteness for stably computable formal systems

本文证明稳定可计算枚举形式系统满足哥德尔第一和第二不完全性定理的类比，并应用于人类数学输出的极限，得出更尖锐的哥德尔析取版本。

关于同胚下[0,1]的计算可枚举闭子集的一个反例

A Counterexample Regarding C.E. Closed Subsets of [0,1] Under Homeomorphisms

该研究构造了[0,1]上一个不与任何计算紧空间同胚的计算可枚举闭子集反例，解决了Koh、Melnikov和Ng提出的相关问题。

Measuring the complexity of reductions between equivalence relations

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable…

On the existence of a strong minimal pair

We show that there is a strong minimal pair in the computably enumerable Turing degrees.

Co-c.e. spheres and cells in computable metric spaces

We investigate conditions under which a co-computably enumerable set in a computable metric space is computable. Using higher-dimensional chains and spherical chains we prove that in each computable metric space which is locally computable each co-computably enumerable sphere is computable and each co-c.e. cell with co-c.e. boundary sphere is computable.

Compact manifolds with computable boundaries

We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with computable boundary is computable. In fact, we examine the notion of a semi-computable compact set and we prove a more general result: in any computable metric space each semi-computable compact manifold with computable boundary is computable. In particular, each semi-computable compact (boundaryless) manifold is computable.

$\u4e2aC^*$-代数的可计算$K$-理论：UHF代数

Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras

本研究开启$\u4e2aC^*$-代数$K$-理论的有效内容研究，证明可计算函子可关联代数与其$K$-群的可计算枚举表示，在UHF代数情形下得到更强的可计算性结果及等价刻画，并证明UHF代数的可计算范畴性。

Learning Theory in the Arithmetic Hierarchy

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the $Σ_5^0$-completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a $Δ_2^0$ enumeration witnessing failure.

Calibrating word problems of groups via the complexity of equivalence relations

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

等价关系的可计算归约性与有效跳跃算子

Computable reducibility of equivalence relations and an effective jump operator

本文提出可计算FS跳跃算子，证明其关于可计算归约性是严格的，并研究该算子对可计算枚举等价关系的影响。

可计算枚举集的宽容学习层次结构

The tolerant learning hierarchy for c.e. sets

研究可计算枚举集的宽容学习层次结构，通过对其索引上等价关系确定正确性，推广经典学习方法，得到新度结构，分析其性质及与基准关系的学习能力，区分非有效与可计算宽容学习，产生了算法学习与等价关系理论接口处的度理论。

On functors enumerating structures

We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the equivalence relation computable are equivalent. We also obtain results on the relation between enumerable and computable functors.

可计算枚举代数类上的同构关系

Isomorphism relations on classes of c.e. algebras

研究可计算枚举代数类的同构关系复杂性，通过可计算约简分析其算法复杂性与代数结构的相互作用。

A note on the category of equivalence relations

We make some beginning observations about the category $\mathbb{E}\mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $\mathbb{E}\mathrm{q}$, such as the category $\mathbb{E}\mathrm{q}(Σ^0_1)$ of computably enumerable equivalence relations (called ceers), the category $\mathbb{E}\mathrm{q}(Π^0_1)$ of co-computably enumerable equivalence relations, a…

Classifying equivalence relations in the Ershov hierarchy

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $Δ^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $\leq_c$ on the $Σ^{-1}_{a}\smallsetminus Π^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existe…

可计算顺从群的可计算Følner序列

Computable Folner sequences of amenable groups

本文研究了可计算枚举顺从群中的可计算Følner序列，将存在性基本结果扩展至非有限生成情形，并探讨了序列族复杂性与向度量群的推广。

The hierarchy of equivalence relations on the natural numbers under computable reducibility

The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e.\ sets, and more specifically, on various classes of c.e.\ structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e.\ graphs or on…

Computable quotient presentations of models of arithmetic and set theory

We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No $Σ_1$-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language $\{+,\cdot,\leq\}$ has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theori…

On the structure of computable reducibility on equivalence relations of natural numbers

We examine the degree structure $\mathbf{ER}$ of equivalence relations on $ω$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\mathbf{ER}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf{d}\oplus \mathbf…

Punctual等价关系及其（punctual）复杂度

Punctual equivalence relations and their (punctual) complexity

本研究探究punctual等价关系上由原始递归归约生成的$\u2009Peq$度结构，证明其为稠密分配格且兼具非刚性、非齐次性，展现出比已知等价关系度结构更丰富的结构与复杂性。

可计算归约背景下可数 Borel 等价关系的类似物

Analogues of the countable Borel equivalence relations in the setting of computable reducibility

本文研究可计算归约下可数Borel等价关系的两种类似物，证明基于Feldman-Moore有效化的概念存在严格二分法，而基于Lusin-Novikov的概念则不存在，并存在介于相等与几乎相等关系之间的链和反链，从而否定了Glimm-Efros二分法的一般类似物。

On permutations with decidable cycles

Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $\mathrm{Perm}$, the group of all recursive permutations $\mathcal{G}$. Multiple equivalent computable representations of decidable equivalence relations are provided. $\mathcal{G}$-conjugacy in $\mathrm{Perm}$ is characterised by computable isomorphy of cycle equivalence relations. This result parallels the equivalence of cycle type equality and conjugacy in the full symmetric group of the natural numbers. Conditions are presented for a permutation $f \i…

可度量图化等价关系

Borel graphable equivalence relations

本文研究了哪些分析等价关系是Borel图化的，并探讨了图群及其结构性质。