We examine the sequences A that are low for dimension, i.e. those for which the effective dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf (...) random sequence has effective dimension 1 relative to A, and lowishness for K, namely, that the limit of KA/K is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0. (shrink)

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a (...) structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well. (shrink)

It is shown that there exist subsets A and B of the real line which are recursively constructible such that A has a nonrecursive Hausdorff dimension and B has a recursive Hausdorff dimension but has a finite, nonrecursive Hausdorff measure. It is also shown that there exists a polynomial-time computable curve on the two-dimensional plane that has a nonrecursive Hausdorff dimension between 1 and 2. Computability of Julia sets of computable functions on the (...) real line is investigated. It is shown that there exists a polynomial-time computable function f on the real line whose Julia set is not recurisvely approximable. (shrink)

In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin‐Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a (...) new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension, the constructive packing dimension, and their generalizations, the constructive scaled dimension and the constructive scaled strong dimension (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from existing (...) MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated. (shrink)

Classical fractal dimensions have recently been effectivized by characterizing them in terms of real-valued functions called gales, and imposing computability and complexity constraints on these gales. This paper surveys these developments and their applications in algorithmic information theory and computational complexity theory.

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10, 3, 7] that every (...) real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one). (shrink)

This paper continues the study of the metric topology on $2^{\mathbb {N}}$ that was introduced by S. Binns. This topology is induced by a directional metric where the distance from $Y\in2^{\mathbb {N}}$ to $X\in2^{\mathbb {N}}$ is given by \[\limsup_{n}\frac{C(X\upharpoonright n|Y\upharpoonright n)}{n}.\] This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on $2^{\mathbb {N}}$ and that under it the functions $X\mapsto\operatorname{dim}_{\mathcal{H}}X$ and $X\mapsto\operatorname{dim}_{p}X$ are continuous. We also (...) investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real $X\in2^{\mathbb {N}}$ by an element $\alpha\in[0,1]$ represents a dilution of the information in $X$ by a factor of $\alpha$ . Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed. (shrink)

Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$.(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every (...) set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where $\mathrm {Dim_P}$ is the packing dimension. (shrink)

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 dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension.We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable (...) subsets. We demonstrate how this notion of $\Pi ^0_1$ -immunity is connected to other immunity notions, and construct $\Pi ^0_1$ -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a $\Pi ^0_1$ -immune set.Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random.Abstract prepared by David J. Webb.E-mail: [email protected]: https://arxiv.org/pdf/2209.05659.pdf. (shrink)

An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ (...) such that (ℝ, +, ·, E) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion $\germ{R}$ of (ℝ, +, ·), every subset of ℝ definable in ( $\germ{R}$ , E) either has interior or is Hausdorff null. (shrink)

We show that if a real x2ω is strongly Hausdorff -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective continuous transformations and a basis theorem for -classes applied to closed sets of probability measures. We use the main result to (...) derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. (shrink)

It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and propose a general method (...) for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. (shrink)

We examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information iswhereK is the prefix-free Kolmogorov complexity. A realAis said to have finite self-information ifI is finite. We give a construction for a perfect Π10class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals withK≤+KA+f for any given Δ20fwith a particularly nice approximation and for a specific choice of (...) f it can also be used to produce a perfect Π10set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfyK≤+KA+f for allfin a ‘nice’ class of Δ20functions which includes all Δ20orders. (shrink)

If is a random sequence, then the sequence is clearly not random; however, seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω and halting self-similar sets, Hokkaido Math. J. 31 219–253] have studied the degree of randomness of sequences or reals by measuring (...) their “degree of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension in terms of strong Martin-Löf ε-tests , and we show that ε-randomness for ε is different than the classical 1-randomness. (shrink)

We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that (...) the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left$ is rational. (shrink)

This paper develops a general theory of uniform probability for compact metric spaces. Special cases of uniform probability include Lebesgue measure, the volume element on a Riemannian manifold, Haar measure, and various fractal measures (all suitably normalized). This paper first appeared fall of 1990 in the Journal of Theoretical Probability, vol. 3, no. 4, pp. 611—626. The key words by which this article was indexed were: ε-capacity, weak convergence, uniform probability, Hausdorff dimension, and capacity dimension.

There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions of weakness. In particular, (...) we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems. (shrink)

We introduce a notion of description for infinite sequences and their sets, and a corresponding notion of complexity. We show that for strict process machines, complexity of a sequence or of a subset of Cantor space is equal to its effective Hausdorff dimension.

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every (...) point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself. (shrink)

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier (...) rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k ≥ k (K,K) = Lev ≥ k+1 (K,K), then Lev ≥ k (K,K) = Lev ≥ω (K,K). A space X ∈ K is co-existentially closed in K if Lev ≥ 0 (K, X) = Lev ≥ 1 (K, X). Adapting the technique of "adding roots," by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one. (shrink)

This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and (...) clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility.We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $\mathbb {N}$ or is single-valued.These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $\mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $\mathsf {DS}$, and the Weihrauch equivalent problem $\mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level.Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $\mathbf {K}$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol {\Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf {K}$. We also generalize the results by relaxing the compactness of the ambient space and show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}$ with the Fell topology. A similar result holds also for the Vietoris topology.We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.prepared by Manlio Valenti.E-mail:[email protected]. (shrink)

Consider a randomness notion C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}. A uniform test in the sense of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} is a total computable procedure that each oracle X produces a test relative to X in the sense of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}. We say that a binary sequence Y is C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}-random uniformly relative to (...) X if Y passes all uniform C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} tests relative to X. Suppose now we have a pair of randomness notions C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} and D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} where C⊆D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C} \subseteq \mathcal{D}}$$\end{document}, for instance Martin-Löf randomness and Schnorr randomness. Several authors have characterized classes of the form Low which consist of the oracles X that are so feeble that C⊆DX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C} \subseteq \mathcal{D}^X}$$\end{document}. Our goal is to do the same when the randomness notion D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} is relativized uniformly: denote by Low \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\star}$$\end{document} the class of oracles X such that every C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}-random is uniformly D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}-random relative to X. We show that X∈Low⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\in Low ^\star}$$\end{document} if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Löf packing measure zero with respect to all computable dimension functions. We also show that X∈Low⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\in Low^\star}$$\end{document} if and only if X is computably i.o. tt-traceable if and only if X is not totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all computable dimension functions. (shrink)

We study concepts of decidability for subsets of Euclidean spaces ℝk within the framework of approximate computability . A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of (...) the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces. (shrink)

Expressives like damn and bastard have, when uttered, an immediate and powerful impact on the context. They are performative, often destructively so. They are revealing of the perspective from which the utterance is made, and they can have a dramatic impact on how current and future utterances are perceived. This, despite the fact that speakers are invariably hard-pressed to articulate what they mean. I develop a general theory of these volatile, indispensable meanings. The theory is built around a class of (...) expressive indices. These determine the expressive setting of the context of interpretation. Expressives morphemes act on that context, actively changing its expressive setting. The theory is multidimensional in the sense that descriptives and expressives are fundamentally different but receive a unified logical treatment. (shrink)

Personal values have long been associated with individual decision behavior. The role played by personal values in decision making within an organization is less clear. Past research has found that managers tend to respond to ethical dilemmas situationally. This study examines the relationship between personal values and the ethical dimension of decision making using Partial Least Squares (PLS) analysis. The study examines personal values as they relate to five types of ethical dilemmas. We found a significant positive contribution of (...) altruistic values to ethical decision making and a significant negative contribution of self-enhancement values to ethical decision making. (shrink)

This paper argues that an adequate conception of a good life should recognize, in addition to happiness and morality, a third dimension of meaningfulness. It further proposes that we understand meaningfulness as involving both a subjective and an objective condition, suitably linked. Meaning arises when subjective attraction meets objective attractiveness. In other words one’s life is meaningful insofar as one is gripped or excited by things worthy of one’s love, and one is able to do something positive about it. (...) The paper concludes with some speculations about how this conception of meaningfulness might help to explain the conditions under which social volunteering can be especially rewarding. (shrink)

We draw on the behavioral agency model to explore the ethical consequences of CEO equity incentives. We argue that CEOs are more concerned with funding pension plans when they have more to gain from their stock options yet will increasingly underfund employee pension funds as their current option wealth increases. Our findings reveal that both effects hold when the CEO has greater power (also occupying board chair) over firm decision making. Our study suggests that there is an ethical dimension (...) to equity incentives, given they are intended to align CEO interests with shareholders, yet potentially incentivize CEO behaviors with adverse consequences for employees. Insights from our findings provide boards and regulators with behavioral levers to protect employee well-being in the context of pension funding. (shrink)

Die These, dass neben dem Wohlergehen und der Moral der Sinn eine dritte, irreduzibel eigenständige und fundamentale Dimension des guten Lebens darstellt, hat im Zuge des erneuten Interesses an der Kategorie des Sinns breite Beachtung gefunden. Das Ziel dieses Beitrags besteht erstens darin zu erkunden, was diese These eigentlich beinhaltet. Bei näherer Prüfung zeigt sich nämlich, dass die These auch und gerade von ihren Vertretern bisher überwiegend durch Appelle an Intuitionen und durch Abgrenzung zu alternativen Positionen, aber eben nur (...) unzureichend positiv motiviert und konturiert wurde. Insbesondere wird vorgeschlagen, zwischen einer axiologischen und einer rationalitätstheoretischen Variante der These zu unterscheiden. Während Sinn gemäß der axiologischen Eigenständigkeitsthese eine irreduzibel eigenständige, fundamentale Wertdimension bildet, entlang derer ein Leben als mehr oder weniger gut qualifiziert werden kann, behauptet die rationalitätstheoretische Eigenständigkeitsthese, dass Sinn eine irreduzibel eigenständige, fundamentale Quelle praktischer Gründe bildet. Diese Unterscheidung ist nicht zuletzt deshalb bedeutsam, weil die zweite These die erste vorauszusetzen scheint, während das Umgekehrte nicht gilt: Selbst Hauptvertreter der axiologischen Eigenständigkeitsthese machen sich die rationalitätstheoretische Variante nur mit großen Einschränkungen zu eigen, die wiederum zu kritischen Fragen auch an die axiologische Fassung Anlass geben. Zweitens soll im Zuge einer Erkundung des durch die axiologische wie auch durch die rationalitätstheoretische Version der Eigenständigkeitsthese aufgespannten Problemfeldes gezeigt werden, dass bisher keine überzeugenden Argumente für die Eigenständigkeitsthese vorgebracht und auch keine plausiblen inhaltlichen Kandidaten für eine Theorie des Sinns, die die Anforderungen an eine solche eigenständige Dimension des guten Lebens und Quelle praktischer Gründe sui generis erfüllen würden, identifiziert werden konnten. Des Weiteren sprechen auch allgemeine wert- und rationalitätstheoretische Erwägungen gegen die Annahme einer solchen Dimension. (shrink)

Planetary protection is not just a matter of science. It is also a matter of value. This is so independently of whether we only include the protection of science or if we also include other goals. Excluding other values than the protection of science is thus a value statement, not a scientific statement and it does not make planetary protection value neutral. It just makes the axiological basis (that is, the value basis) for planetary protection more limited in a way (...) that is inconsistent with the axiological grounds for back contamination, ethically questionable and strategically unwise. However we look at it, we cannot get away from the conclusion that the axiological dimension of planetary protection is a task that needs to involve experts on value theory as well as experts from a range of different sciences and also include opinions from outside the academic community. (shrink)

Hersch’s objective in his paper was to “illustrate how an existential ontology has a great deal to offer psychotherapists”. The first of three sections addresses existential themes such as guilt and anxiety and explores the notion of bad faith; the second focuses on why existential ontology provides a more suitable grounding for psychotherapy than traditional models; and the third offers the author’s invention of a mental status examination that is derived from existential ontology. To illustrate how existential ontology may be (...) useful to psychotherapists, Hersch directs our attention to those existential themes that are crucial to understanding the role ontology plays, existential.. (shrink)

PART I The superpower and asymmetry PART II Jus ad bellum, jus in bello, jus post bellum PART III Leadership and accountability PART IV Soldiersa (TM) ...

In Locke’s economic and politic thinking is possible appraise some articulations that connect the teological, the economic and the politic. The humans work for divine decree and the labor is an economic concept inasmuch as it is the activity that yields private ownership, whose possession entails a natural right for all individuals. Safeguarding this right is liability of the government and of the state institutions. For Locke the market is previous to the civilian society, trade’s regulative structure sets the guidelines (...) of the governmental action, so that money’s circulation should not be interfered directly by the sovereign power neither leading it under particular politic interests. Thus, it is can find out an ontology of market in lockean thought, insomuch as trade space’s existence takes up the natural laws and project them in the space`s political. (shrink)

Looks at the ways people soothe or console themselves when faced with suffering and studies the psychological use of transitional objects.

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.