arXiv subjects

Wolfgang Merkle

Publications and source records attributed to Wolfgang Merkle.

8 recordsLinked to original sources

arXiv2026

Speedability of computably approximable reals and their approximations

An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.

math.LO↗

arXiv2024

Randomness versus superspeedability

Speedable numbers are real numbers which are algorithmically approximable from below and whose approximations can be accelerated nonuniformly. We begin this article by answering a question of Barmpalias by separating a strict subclass that we will refer to as superspeedable from the speedable numbers; for elements of this subclass, acceleration is possible uniformly and to an even higher degree. This new type of benign left-approximations of numbers then integrates itself into a hierarchy of other such notions studied in a growing body of recent work. We add a new perspective to this study by juxtaposing this hierachy with the well-studied hierachy of algorithmic randomness notions.

math.LO↗

arXiv2024

A total Solovay reducibility and totalizing of the notion of speedability

While the set of Martin-Löf random left-c.e. reals is equal to the maximum degree of Solovay reducibility, Miyabe, Nies and Stephan(DOI:10.4115/jla.2018.10.3) have shown that the left-c.e. Schnorr random reals are not closed upwards under Solovay reducibility. Recall that for two left-c.e. reals~$α$ and~$β$, the former is Solovay reducible to the latter in case there is a partially computable function $φ$ and constant~$c$ such that for all rational numbers $q < β$ we have \[α- φ(q) < c(β- q).\] By requiring the translation function $φ$ to be total, we introduce a total version of Solovay reducibility that implies Schnorr reducibility. Accordingly, by Downey and Griffiths (DOI:10.2178/jsl/1082418542), the set of Schnorr random left-c.e. reals is closed upwards relative to total Solovay reducibility. Furthermore, we observe that the notion of speedability introduced by Merkle and Titov (DOI:10.1007/978-3-030-50026-9_22) can be equivalently characterized via partial computable translation functions in a way that resembles Solovay reducibility. By requiring the translation function to be total, we obtain the concept of total speedability. Like for speedability, this notion does not depend on the choice of the speeding constant.

math.LO↗

arXiv2021

Relativized depth

Bennett's notion of depth is usually considered to describe the usefulness and internal organization of the information encoded into an object such as an infinite binary sequence. We consider a natural way to relativize the notion of depth for such sets, and we investigate for various kinds of oracles whether and how the unrelativized and the relativized version of depth differ. Intuitively speaking, access to an oracle increases computation power. Accordingly, for most notions for sets considered in computability theory, for the corresponding classes trivially for all oracles the unrelativized class is contained in the relativized class or for all oracles the relativized class is contained in the unrelativized class. Examples for these two cases are given by the classes of computable and of Martin-Löf random sets, respectively. However, in the case for depth the situation is different. It turns out that the classes of deep sets and of sets that are deep relative to the halting set $\emptyset '$ are incomparable with respect to set-theoretical inclusion. On the other hand, the class of deep sets is strictly contained in the class of sets that are deep relative to any given Martin-Löf-random oracle. The set built in the proof of the latter result can also be used to give a short proof of the known fact that every PA-complete degree is Turing-equivalent to the join of two Martin-Löf-random sets. In fact, we slightly strengthen this result by showing that every DNC$_2$ function is truth-table-equivalent to the join of two Martin-Löf random sets. Furthermore, we observe that the class of deep sets relative to any given K-trivial oracle either is the same as or is strictly contained in the class of deep sets. Obviously, the former case applies to computable oracles. We leave it as an open problem which of the two possibilities can occur for noncomputable K-trivial oracles.

math.LO↗

arXiv2019

Normalized Information Distance and the Oscillation Hierarchy

We study the complexity of approximations to the normalized information distance. We introduce a hierarchy of computable approximations by considering the number of oscillations. This is a function version of the difference hierarchy for sets. We show that the normalized information distance is not in any level of this hierarchy, strengthening previous nonapproximability results. As an ingredient to the proof, we also prove a conditional undecidability result about independence.

math.LO↗

arXiv2016

Solovay functions and their applications in algorithmic randomness

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.

math.LO↗

arXiv2014

Kolmogorov complexity and the Recursion Theorem

Several classes of DNR functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0,1}-valued DNR function, iff A can compute a function F such that F(n) is a string of length n and maximal C-complexity among the strings of length n. A solves the halting problem iff A can compute a function F such that F(n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem.

math.LO↗

arXiv2009

Separations of non-monotonic randomness notions

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-Loef randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of Martin-Loef randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr's model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent ``non-monotonic'' notion of randomness, now called Kolmogorov-Loveland randomness, has been shown to be quite close to Martin-Loef randomness, but whether these two classes coincide remains a fundamental open question. As suggested by Miller and Nies, we study in this paper weak versions of Kolmogorov-Loveland randomness, where the betting strategies are non-adaptive (i.e., the positions of the bits to bet on should be decided before the game). We obtain a full classification of the different notions we consider.

cs.CC↗