arXiv subjects
Publications and source records attributed to André Nies.
We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. In the process, we obtain the result of independent interest that the epimorphism relation on pointed reflexive graphs is complete.
We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a bounded difference is Borel reducible to it. This marks substantial progress on an open problem of Kechris, Nies and Tent (2018): to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups.
The Jordan decomposition theorem states that every function $f \colon [0,1] \to \mathbb{R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function of bounded variation is differentiable almost everywhere in the sense of Lebesgue measure. We analyze the strength of these theorems in the setting of reverse mathematics. Over $\mathsf{RCA}_0$, a stronger version of Jordan's result where all functions are continuous is equivalent to $\mathsf{ACA}_0$, while the version stated is equivalent to $\mathsf{WKL}_0$. The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to $\mathsf{WWKL}_0$. To state this equivalence in a meaningful way, we develop a theory of Martin-Löf randomness over $\mathsf{RCA}_0$.
For $p \in [0,1]$ let $\mathcal D(p)$ be the mass problem of infinite bit sequences~$y$ (i.e., $\{0,1\}$-valued functions) such that for each computable bit sequence $x$, the bit sequence $ x \leftrightarrow y$ has asymptotic lower density at most $p$ (where $x \leftrightarrow y$ has a $1$ in position $n$ iff $x(n) = y(n)$). We show that all members of this family of mass problems parameterized by a real $p$ with $0 < p<1/2 $ have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems $\mathcal D(p)$ with the mass problem $\mathrm{IOE}(2^ { 2^ n})$. As a dual of the problem $\mathcal D(p)$, define $\mathcal B(p)$, for $0 \le p < 1/2$, to be the set of bit sequences $y$ such that $\underline ρ(x \leftrightarrow y) > p$ for each computable set~$x$. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems $\mathcal B(p)$ is the same for all $p \in (0, 1/2)$, by showing that they are Medvedev equivalent to the mass problem of functions bounded by $2^{2^ n}$ that are almost everywhere different from each computable function. Together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type $\mathrm{IOE}$: We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above.
A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form SNR$_f$ consisting of SNR functions bounded by varying recursive bounds $f$. We show that the connection between SNR functions and canonically immune sets is, in a sense, as strong as that between DNR (diagonally non-recursive) functions and effectively immune sets. Finally, we introduce pandemic numberings, a set-theoretic dual to immunity.
We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between $2$-randomness and being infinitely often $C$-incompressible is provable in $\mathsf{RCA}_0$. We verify that $\mathsf{RCA}_0$ proves the basic implications among randomness notions: $2$-random $\Rightarrow$ weakly $2$-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $\mathsf{RCA}_0$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.
The general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.
We extend the key notion of Martin-Löf randomness for infinite bit sequences to the quantum setting, where the sequences become states of an infinite dimensional system. We work towards showing an analogy with the Levin-Schnorr theorem to characterise quantum ML-randomness of states by incompressibility (in the sense of quantum Turing machines) of all initial segments.
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points. We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.
(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.
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.
Classical versions of Kolmogorov complexity are incomputable. Nevertheless, in 1975 Solovay showed that there are computable functions $f > K+O(1)$ such that for infinitely many strings $σ$, $f(σ)=K(σ)+O(1)$, where $K$ denotes prefix-free Kolmogorov complexity (while $C$ denotes plain Kolmogorov complexity). Such an $f$ is now called a Solovay function. We prove that many classical results about $K$ can be obtained by replacing $K$ by a Solovay function. For example, the three following properties of a function $g$ all hold for the function $K$. (i) The sum of the terms $\sum_n 2^{-g(n)}$ is a Martin-Löf random real. (ii) A sequence A is Martin-Löf random if and only if $C(A \upharpoonright n) > n -g(n)-O(1)$. (iii) A sequence A is K-trivial if and only if $K(A \upharpoonright n) < g(n) + O(1)$. We show that when fixing any of these three properties, then among all computable functions exactly the Solovay functions possess this property. Furthermore, this characterization extends accordingly to the larger class of right-c.e. functions.
We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than \ML\ (ML) randomness. We establish several equivalences. Given a ML-random real $z$, the additional randomness strengths needed for the following are equivalent. \n (1) all effectively closed classes containing $z$ have density $1$ at $z$. \n (2) all nondecreasing functions with uniformly left-c.e.\ increments are differentiable at $z$. \n (3) $z$ is a Lebesgue point of each lower semicomputable integrable function. We also consider convergence of left-c.e.\ martingales, and convergence in the sense of Birkhoff's pointwise ergodic theorem. Lastly we study randomness notions for density of $Π^0_n$ and $Σ^1_1$ classes.
We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong. While this analogy was first studied explicitly by Rupprecht in his PhD thesis, many prior results can be viewed from this perspective. 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.
A real $x$ is $Δ^1_1$-Kurtz random ($Π^1_1$-Kurtz random) if it is in no closed null $Δ^1_1$ set ($Π^1_1$ set). We show that there is a cone of $Π^1_1$-Kurtz random hyperdegrees. We characterize lowness for $Δ^1_1$-Kurtz randomness as being $Δ^1_1$-dominated and $Δ^1_1$-semi-traceable.
We answer a question of Ambos-Spies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class superhigh$^\Diamond$, and show that it contains some, but not all, of the noncomputable $K$-trivial sets.
Osvald Demuth (1936--1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later. In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth's work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth's independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth's work.