Approximation systems for functions in topological and in metric spaces

A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Two ways to eliminate the using of such names in certain cases are indicated in the paper. The first one is intended for the case of topological spaces with selected indexed denumerable bases. Suppose a partial function is given from one such space into another one whose selected base has a recursively enumerable index set, and suppose that the intersection of base open sets in the first space is computable in the sense of Weihrauch-Grubba. Then the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable relation between indices of base sets containing the argument value and indices of base sets containing the corresponding function value.This result can be regarded as an improvement of a result of Korovina and Kudinov. The second way is applicable to metric spaces with selected indexed denumerable dense subsets. If a partial function is given from one such space into another one, then, under a semi-computability assumption concerning these spaces, the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable set of quadruples. Any of them consists of an index of element from the selected dense subset in the first space, a natural number encoding a rational bound for the distance between this element and the argument value, an index of element from the selected dense subset in the second space and a natural number encoding a rational bound for the distance between this element and the function value. One of the examples in the paper indicates that the computability of real functions can be characterized in a simple way by using the first way of elimination of the infinitistic names.Comment: 21 pages, published in Logical Methods in Computer Scienc

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

Finding subsets of positive measure

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero (and indeed finite) $\mathcal H^s$-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) $\Sigma^1_1$ set of reals in Cantor space, there is always a $\Pi^0_1(\mathcal{O})$ subset on non-zero $\mathcal H^s$-measure definable from Kleene's $\mathcal O$. On the other hand, there are $\Pi^0_2$ sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The Strength of the Besicovitch--Davies Theorem". The final publication of that paper is available at Springer via http://dx.doi.org/10.1007/978-3-642-13962-8_2

Computability questions in the sphere packing problem

We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.Comment: 24 pages, LaTe

On the Continuity of Effective Multifunctions

AbstractIf one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed in better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) in (the code of) a description for the generation of the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest and well known theorems such as those of Myhill and Shepherdson, Kreisel, Lacombe and Shoenfield, Ceĭtin, and/or Moschovakis present answers for important special cases. A general, unifying approach has been developed by the present author in [D. Spreen. On effective topological spaces. The Journal of Symbolic Logic, 63 (1998), 185–221. Corrections ibid., 65 (2000), 1917–1918].In this paper the approach is extended to the case of multifunctions. Such functions appear very naturally in applied mathematics, logic and theoretical computer science. Various ways of coding (indexing) sets are discussed and effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system and possess certain witness functions are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Important special cases are discussed where such witnessing functions always exist

Effective Dispersion in Computable Metric Spaces

We investigate the relationship between computable metric spaces $(X,d,alpha )$ and $(X,d,beta ),$ where $(X,d)$ is a given metric space. In the case of Euclidean space, $alpha$ and $beta$ are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space. This notion is essential in the proof of the main result of this paper: $(X,d,alpha )$ is effectively totally bounded if and only if $(X,d,beta )$ is effectively totally bounded, i.e. the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space

Effectively closed sets and graphs of computable real functions

AbstractIn this paper, we compare the computability and complexity of a continuous real function F with the computability and complexity of the graph G of the function F. A similar analysis will be carried out for functions on subspaces of the real line such as the Cantor space, the Baire space and the unit interval. In particular, we define four basic types of effectively closed sets C depending on whether (i) the set of closed intervals which with nonempty intersection with C is recursively enumerable (r.e.), (ii) the set of closed intervals with empty intersection with C is r.e., (iii) the set of open intervals which with nonempty intersection with C is r.e., and (iv) the set of open intervals with empty intersection with C is r.e. We study the relationships between these four types of effectively closed sets in general and the relationships between these four types of effectively closed sets for closed sets which are graphs of continuous functions