Contemporary scientific studies frequently rely on data-intensive analytical computing. While the main goal of this emerging form of computing is to facilitate hypothesis formulation or to test the validity of a postulated model, its primary method is usually that of data clustering. Since typical analytical tasks operate on very large volumes of potentially highdimensional data, scientific stu...

The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆2-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆2-Turing degree contains a d-c.e. real.

This is a report for my presentation at the upcoming meeting on Berechenbarkeitstheorie (“Computability Theory”), Oberwolfach, January 21–27, 2001. We use 2 to denote the space of infinite sequences of 0’s and 1’s. For X, Y ∈ 2, X ≤T Y means that X is Turing reducible to Y . For P,Q ⊆ 2 we say that P is Muchnik reducible to Q, abbreviated P ≤w Q, if for all Y ∈ Q there exists X ∈ P such that X ...

A set A is coarsely computable with density r ∈ [0,1] if there is an algorithm for deciding membership in A which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least r. To any Turing degree a we can assign a value ΓT (a): the minimum, over all sets A in a, of the highest density at which A is coarsely computable. The closer ΓT (a) is to 1, the clos...

When creating datasets for modelling of human skills based on training examples from human motion, one can encounter the problem that the kinematics of the robot does not match the human kinematics. Presented is a simple method of bypassing the explicit modelling of the human kinematics based on a variant of the self-organising map (SOM) algorithm. While the literature contains instances of SOM...

Introduction* Let Mj denote the general decision problem for Markov algorithms with axiom. Of interest to us is whether or not this class of problems is as richly structured, with regard to degrees of unsolvability, as those classes studied in Hughes, Overbeek, and Singletary [2]. In this paper we shall present proofs which show this to be so. In particular we shall show that the general decisi...

1. Introduction. A decision problem for a combinatorial system shall denote a pair ($, S) where is a specified kind of decision problem (e.g. derivability problem, halting problem, etc.) and 5 is a combina-torial system. Two decision problems (<£i, Si), ($ 2 , £2) are sa ^d t0 be of the same many-one degree (of unsolvability) if there exist effective many-one mappings ƒ and g such that each...

Games are an important tool in mathematics and logic, providing a clear and intuitive understanding of the notions they define or characterize. In particular, since the seminal work of Wadge in the 1970s, game characterizations of classes of functions in Baire space have been a rich area of research, having had significant and far-reaching development by van Wesep, Andretta, Duparc, Motto Ros, ...

The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(a), that is, p is majorised by q. For X being hyperi...

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we determine the exact value s(T ) for trees T in which every two v...