A Galois connection between Turing jumps and limits

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis

Algorithmic Randomness

We consider algorithmic randomness in the Cantor space C of the infinite binary sequences. By an algorithmic randomness concept one specifies a set of elements of C, each of which is assigned the property of being random. Miscellaneous notions from computability theory are used in the definitions of randomness concepts that are essentially rooted in the following three intuitive randomness requirements: the initial segments of a random sequence should be effectively incompressible, no random sequence should be an element of an effective measure null set containing sequences with an “exceptional property”, and finally, considering betting games, in which the bits of a sequence are guessed successively, there should be no effective betting strategy that helps a player win an unbounded amount of capital on a random sequence. For various formalizations of these requirements one uses versions of Kolmogorov complexity, of tests, and of martingales, respectively. In case any of these notions is used in the definition of a randomness concept, one may ask in general for fundamental equivalent definitions in terms of the respective other two notions. This was a long-standing open question w.r.t. computable randomness, a central concept that had been introduced by Schnorr via martingales. In this thesis, we introduce bounded tests that we use to give a characterization of computable randomness in terms of tests. Our result was obtained independently of the prior test characterization of computable randomness due to Downey, Griffiths, and LaForte, who defined graded tests for their result. Based on bounded tests, we define bounded machines which give rise to a version of Kolmogorov complexity that we use to prove another characterization of computable randomness. This result, as in analog situations, allows for the introduction of interesting lowness and triviality properties that are, roughly speaking, “anti-randomness” properties. We define and study the notions lowness for bounded machines and bounded triviality. Using a theorem due to Nies, it can be shown that only the computable sequences are low for bounded machines. Further we show some interesting properties of bounded machines, and we demonstrate that every boundedly trivial sequence is K-trivial. Furthermore we define lowness for computable machines, a lowness notion in the setting of Schnorr randomness. We prove that a sequence is low for computable machines if and only if it is computably traceable. Gacs and independently Kucera proved a central theorem which states that every sequence is effectively decodable from a suitable Martin-Löf random sequence. We present a somewhat easier proof of this theorem, where we construct a sequence with the required property by diagonalizing against appropriate martingales. By a variant of that construction we prove that there exists a computably random sequence that is weak truth-table autoreducible. Further, we show that a sequence is computably enumerable self-reducible if and only if its associated real is computably enumerable. Finally we investigate interrelations between the Lebesgue measure and effective measures on C. We prove the following extension of a result due to Book, Lutz, and Wagner: A union of Pi-0-1 classes that is closed under finite variations has Lebesgue measure zero if and only if it contains no Kurtz random real. However we demonstrate that even a Sigma-0-2 class with Lebesgue measure zero need not be a Kurtz null class. Turning to Almost classes, we show among other things that every Almost class with respect to a bounded reducibility has computable packing dimension zero

Levels of discontinuity, limit-computability, and jump operators

We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $\Delta^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem

Computability and Fractal Dimension

This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000). Working in the Cantor space of all infinite binary sequences, we study the theory of Hausdorff and other dimensions for individual sequences. After giving an overview over the classical theory of fractal dimension in Cantor space, we develop the theory of effective Hausdorff dimension and its variants systematically. Our presentation is inspired by the approach to algorithmic information theory developed by Kolmogorov and his students. We are able to give a new and much easier proof of a central result of the effective theory: Effective Hausdorff dimension coincides with the lower asymptotic algorithmic entropy, defined in terms of Kolmogorov complexity. Besides, we prove a general theorem on the behavior of effective dimension under r-expansive mappings, which can be seen as a generalization of Hölder mappings in Cantor space. Furthermore, we study the connections between other notions of effective fractal dimension and algorithmic entropy. Besides, we are able to show that the set of sequences of effective Hausdorff dimension s has Hausdorff dimension s and infinite s-dimensional Hausdorff measure (for every 0<s<1). Next, we study the Hausdorff dimension (effective and classical) of objects arising in computability theory. We prove that the upper cone of any sequence under a standard reducibility has Hausdorff dimension 1, thereby exposing a Lebesgue nullset that has maximal Hausdorff dimension. Furthermore, using the behavior of effective dimension under r-expansive transformations, we are able to show that the effective Hausdorff dimension of the lower cone and the degree of a sequence coincide. For many-one reducibility, we prove the existence of lower cones of non-integral dimension. After giving some natural' examples of sequences of effective dimension 0, we prove that every effectively closed set A of positive Hausdorff dimension admits a computable, surjective mapping onto Cantor space. We go on to study the complex interrelation between algorithmic entropy, randomness, effective Hausdorff dimension, and reducibility more closely. For this purpose we generalize effective Hausdorff dimension by introducing the notion of strong effective Hausdorff measure 0. We are able to show that not having strong effective Hausdorff measure 0 does not necessarily allow to compute a Martin-Löf random sequence, a sequence of highest possible algorithmic entropy. Besides, we show that a generalization of the notion of effective randomness to noncomputable measures yields a very coarse concept of randomness in the sense that every noncomputable sequence is random with respect to some measure. Next, we introduce Schnorr dimension, a notion of dimension which is algorithmically more restrictive than effective dimension. We prove a machine characterization of Schnorr dimension and show that, on the computably enumerable sets, Schnorr Hausdorff dimension and Schnorr packing dimension do not coincide, in contrast to the case of effective dimension. We also study subrecursive notions of effective Hausdorff dimension. Using resource-bounded martingales, we are able to transfer the use of r-expansiveness to the resource-bounded case, which enables us to show that the Small-Span Theorem does not hold for dimension in exponential time E. Finally, we investigate the effective Hausdorff dimension of sequences against which no computable nonmonotonic betting strategy can succeed. Computable nonmonotonic betting games are a generalization of computable martingales, and it is a major open question whether the randomness notion induced by them is equivalent to Martin-Löf randomness. We are able to show that the sequences which are random with respect to computable nonmonotonic betting games have effective Hausdorff dimension 1, which implies that, from the viewpoint of algorithmic entropy, they are rather close to Martin-Löf randomness

An Introduction to Mathematical Logic

This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's definability theorem and Herbrand's theorem. It continues with a section on G\"odel's incompleteness theorems, which includes a discussion of first-order arithmetic and primitive recursive functions. This is followed by three sections that are devoted, respectively, to proof theory (provably total recursive functions and Goodstein sequences for $\mathsf{I\Sigma}_1$), computability (fundamental notions and an analysis of K\H{o}nig's lemma in terms of the low basis theorem) and model theory (ultraproducts, chains and the Ax-Grothendieck theorem). We conclude with some brief introductory remarks about set theory (with more details reserved for a separate lecture). The author uses these notes for a first logic course for undergraduates in mathematics, which consists of 28 lectures and 14 exercise sessions of 90 minutes each. In such a course, it may be necessary to omit some material, which is straightforward since all sections except for the first two are independent of each other

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma

We classify the computational content of the Bolzano-Weierstrass Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a computable metric space is the cluster point problem of that space. By specialization to sequences with a relatively compact range we obtain a characterization of the Bolzano-Weierstrass Theorem as the derivative of compact choice. In particular, this shows that the Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the jump of the lesser limited principle of omniscience LLPO and the Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump of the idempotent closure of LLPO. We also introduce the compositional product of two Weihrauch degrees f and g as the supremum of the composition of any two functions below f and g, respectively. We can express the main result such that the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's Lemma and the Monotone Convergence Theorem. We also study the class of weakly limit computable functions, which are functions that can be obtained by composition of weakly computable functions with limit computable functions. We prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this class. Likewise, the unique cluster point problem on real numbers is complete for the class of functions that are limit computable with finitely many mind changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers and, more generally, the unbounded cluster point problem on real numbers is uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias Schr\"oder, and the authors of the original paper. The addendum closes a gap in the proof of Theorem 11.2, which characterizes the computational content of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity