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)

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δta2. Effective difference chains of transfinite (but constructive) order types, consisting of complements of effectively exhaustible sets, as (...) well as another closely related concept, yield a rich hierarchy within the whole class Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS-like (...) manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper. (shrink)

Continuing the paper [7], in which the Blum-Shub-Smale approach to computability over the reals has been generalized to arbitrary algebraic structures, this paper deals with computability and recognizability over structures of infinite signature. It begins with discussing related properties of the linear and scalar real structures and of their discrete counterparts over the natural numbers. Then the existence of universal functions is shown to be equivalent to the effective encodability of the underlying structure. Such structures even have universal functions satisfying (...) the s-m-n theorem and related features. The real and discrete examples are discussed with respect to effective encodability. Megiddo structures and computational extensions of effectively encodable structures are encodable, too. As further variants of universality, universal functions with enumerable sets of program codes and such ones with constructible codes are investigated. Finally, the existence of m-complete sets is shown to be independent of the effective encodability of structures, and the linear and scalar structures are discussed once more, under this aspect. (shrink)

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δta2. Effective difference chains of transfinite order types, consisting of complements of effectively exhaustible sets, as well as (...) another closely related concept, yield a rich hierarchy within the whole class Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (shrink)

I hope you have had no trouble identifying this issue of our journal, even if the cover has been changed slightly. I would like to draw your attention to a more essential improvement in the journal's affairs: namely, that the Editorial Board has been enlarged by four new members. These are the following: Douglas Bridges , Ulrich Kohlenbach , H. Dugald Macpherson , and Pavel Pudlak . Please don't hesitate to contact a fellow editor, in particular the managing editor, if (...) you have suggestions for improving either the editorial procedures or the content and form of MLQ.The aims and scope of our journal are unchanged from the original ones developed by Günter Asser. His name is indissolubly connected with the former “Zeitschrift für Mathematische Logik und Grundlagen der Mathematik ”, now known as “Mathematical Logic Quarterly ”. He was one of the two founding editors and then the editor-in-chief of the journal until the 50th volume appeared about one year ago. Single-handedly, and almost incredibly, he showed that it is possible for one man to be the main editor of a successful highlevel scientific journal throughout half a century. He remains involved in and concerned with the editorial process of MLQ. For example, he carefully looks at every article to be published, faithfully checking almost every line of the final layout.On 26 February 2006, Professor Dr. Günter Asser celebrates his 80th birthday. I would like to take this opportunity, on behalf of the whole readership and of all fellow editors of ZML and MLQ throughout the last 51 years, to congratulate him cordially and to wish him good health and all the best for the future.The present issue starts with a paper directly related to G. Asser's spectrum conjecture. Then there are some papers specially dedicated to him, as well as further contributions, as always scrutinised and then recommended by expert referees. So this turns out to be an appropriate birthday issue. (shrink)

We consider the target-reaching problem in plane scenes for a point robot which has a tactile sensor and can locate the target ray. It might have a compass, too, but it is not able to perceive the coordinates of its position nor to measure distances. The complexity of an algorithm is measured by the number of straight moves until reaching the target, as a function of the number of vertices of the scene. It is shown how the target point can (...) be reached by exhaustive search without using a compass, with the complexity exp). Using a compass, there is a target-reaching algorithm, based on rotation counting, with the complexity O. The decision problem, to recognize if the target cannot be reached because it belongs to an obstacle, cannot be solved by our type of robot. If the behaviour of a robot without compass is periodic in a homogeneous environment, it cannot solve the target-reaching problem. (shrink)

Based on the computation mode introduced in [13], we deal with the time complexity of computations over arbitrary first-order structures.The main emphasis is on parameter-free computations. Some transfer results for solutions of P versus NP problems as well as relationships to quantifier elimination are discussed. By computation tree analysis using first-order formulas, it follows that P versus NP solutions and other results of structural complexity theory are invariant under elementary equivalence of structures.

It is shown that the classes of discrete parts, A ∩ ℕk, of approximately resp. weakly decidable subsets of Euclidean spaces, A ⊆ ℝk, coincide and are equal to the class of ω-r. e. sets which is well-known as the first transfinite level in Ershov's hierarchy exhausting Δ02.

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝ k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta 2. This is based on suitable (...) characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented. (shrink)