相关论文: KL-randomness and effective dimension under strong…

While it is not known whether each real that is Kolmogorov-Loveland random is Martin-L\"of random, i.e., whether $\mathrm{KLR}\subseteq\mathrm{MLR}$, Kjos-Hanssen and Webb (2021) showed that $\mathrm{MLR}$ is truth-table Medvedev reducible…

We prove various results connected together by the common thread of computability theory. First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing…

We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with…

We study Medvedev reducibility in the context of set theory -- specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li \cite{HaLi}, we show that the Medvedev degrees of countable ordinals are far from…

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and…

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr,…

We characterize Martin-L\"of randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus…

In this paper, we study the strong consistency of a bias reduced kernel density estimator and derive a strongly con- sistent Kullback-Leibler divergence (KLD) estimator. As application, we formulate a goodness-of-fit test and an…

In this survey we discuss work of Levin and V'yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V'yugin…

The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x|n)$ (here $\KS(x|n)$ is…

We study finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and…

We survey the theory of Muchnik (weak) and Medvedev (strong) degrees of subsets of ${}^\omega\omega$ with particular attention to the degrees of $\Pi^0_1$ subsets of ${}^\omega2$. Later sections present proofs, some more complete than…

We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table…

We establish isomorphisms between certain specializations of Birman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an…

We study generalizations of Demuth's Theorem, which states that the image of a Martin-L\"of random real under a tt-reduction is either computable or Turing equivalent to a Martin-L\"of random real. We show that Demuth's Theorem holds for…

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table…

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model $\mathcal L \subseteq \mathbb{C}^n$ of dimension $r$ is equal to $(-2)^r\chi_M( \textstyle\frac{1}{2})$, where $\chi_M$ is the…

In this paper, we study a subclass of the class of MD-algebras, i.e., the class of solvable real Lie algebras such that the K-orbits of its corresponding connected and simply connected Lie groups are either orbits of dimension zero or…

Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements in the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical…

This paper provides a unified perspective for the Kullback-Leibler (KL)-divergence and the integral probability metrics (IPMs) from the perspective of maximum likelihood density-ratio estimation (DRE). Both the KL-divergence and the IPMs…