Computability Theory

Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work

Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective

ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on “Synthetic Philosophy of Contemporary Mathematics” by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..

Selection theorem for systems with inheritance

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of "natural" selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t goes to infinity. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of "completely thin" sets is introduced. Key words: Dynamics; Attractor; Evolution; Entropy; Natural selectionComment: 46 pages, the final journal versio

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

Randomness and Computability

This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets