Lowness for isomorphism, countable ideals, and computable traceability

Degrees of and lowness for isometric isomorphism

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces…

Lowness, randomness, and computable analysis

Analytic concepts contribute to our understanding of randomness of reals via algorithmic tests. They also influence the interplay between randomness and lowness notions. We provide a survey, written on the occasion of Rod Downey's 60th birthday.

Taking the path computably travelled

We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $Π^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.

Fixpoints and relative precompleteness

We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings, lowness and relative precompleteness coincide with being computable. We also study the complexity of Skolem functions arising from Arslanov's completeness criterion with parameters. We show that for suitably divisible numberings, these Skolem functions have the maximal possible Turing degree. In particular this holds for the standard numberings of the partial com…

通过开放式覆盖的随机性和低性概念

Randomness and lowness notions via open covers

算法随机性研究的一条主线是低度概念。给定一个随机性概念R，我们问哪些序列A相对于A使R保持不变（即，R^A = R）？这样的序列对于R来说是低的。这个问题延伸到一对随机性概念R和S，其中S更弱：对于哪一个A是S^A仍然弱于R？在过去的几年中，许多结果的特点是低随机性的序列，其低计算强度。一些结果也给出了低序列的测度论特征。例如，Kjos-Hanssen证明了对于Martin-L，A是低的随机性当且仅当每个A-c. e.测度小于1的开集可以用c.e.开测度集小于1。在本文中，我们给出了一系列结果，表明各种各样的低概念可以用类似的方式表达，即，通过某种类型的开集覆盖某种类型的开集的能力。这提供了一个统一的框架，澄清了随机性概念的低研究，并允许我们给出一些已知结果的简单证明。我们还使用这个框架来证明新的结果，包括表明类低（MLR;SR）和低（W2 R;SR）相吻合，回答了Nies的问题。其他的应用包括随机性概念的特征，一个广泛适用的解释，为什么随机性的低与测试的低是相同的，以及一个简单的证明，低（W2 R;S）=低（MLR;S），其中S是马丁-L的类可计算或Schnorr随机序列。最后一节使用可和函数和收敛测度机代替开覆盖给出了低度概念的刻画。最后，我们简单证明Nies的一个结果，即Low（MLR）= Low（MLR; CR）。

强跳可溯性的内在可枚举性

Inherent enumerability of strong jump-traceability

我们证明了每个强跳迹集服从每个良性费用函数。此外，我们还证明了每个强跳迹集都是可从可计算的强跳迹集计算的。这使我们能够概括c.e.的性质。强跳迹集到所有这样的集合。例如，强跳可溯集在图灵度中引入一个理想;强跳可溯集恰好是那些可以从所有超低马丁-洛夫随机集计算的集合;强跳可溯集恰好是那些作为$\text{Demuth}_{\text{BLR}}$ -随机性的基础的集合;强跳可溯等价于强超低性。