Vasco Brattka

Computability on subsets of Euclidean space I: closed and compact subsets

In this paper we introduce and compare computability concepts on the set of closed subsets of Euclidean space. We use the language and framework of Type 2 Theory of Effectivity (TTE) which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems. In particular, Type 2 Theory of Effectivity allows to separate topological from computational aspects of effectivity. We consider three different computability concepts on the set of closed subsets, each of which is characterized by several representations which are proved to be equivalent. The three induced types of computable closed sets have already been considered by many authors, however, under different and partly inconsistent names. Our characterizations show that they can be regarded as straightforward generalizations of the r.e., co-r.e., and recursive subsets of natural numbers. Therefore, we suggest to call them the recursively enumerable, the co-recursively enumerable, and the recursive closed subsets of Euclidean space. Open subsets obtain the dual names. We extend the investigation by introducing several natural representations of the compact subsets of Euclidean space and proving equivalences. The paper extends and generalizes earlier definitions, adds new ones and compares them in a single framework. The resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets.

Effective choice and boundedness principles in computable analysis

In this paper we study a new approach to classify mathemat- ical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially or- dered degree structure induced by it. We have identied certain choice principles on closed sets which are cornerstones among Weihrauch de- grees and it turns out that certain core theorems in analysis can be classied naturally in this structure. In particular, we study theorems

Weihrauch degrees, omniscience principles and weak computability

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnigs Lemma and hence to the Hahn–Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleenes ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense.

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Closed choice and a uniform low basis theorem

Abstract We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, the natural numbers, Cantor space and Baire space correspond to the class of computable functions, functions computable with finitely many mind changes, weakly computable functions and effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. The class of limit computable functions can be characterized with parallelized choice of natural numbers. On top of these results we provide further insights into algebraic properties of closed choice. In particular, we prove that closed choice on Euclidean space can be considered as “locally compact choice” and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be “divided” by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As a main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology.

Recursive characterization of computable real-valued functions and relations

Abstract Corresponding to the definition of μ-recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators. We prove that our class of recursive relations essentially coincides with our class of densely computable relations, defined via Turing machines. In the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real-valued functions, defined via Turing machines by Grzegorczyk, Lacombe and others.

Computability over Topological Structures

Computable analysis is the Turing machine based theory of computability on the real numbers and other topological spaces. Similarly as Ersov’s concept of numberings can be used to deal with discrete structures, Kreitz and Weihrauch’s concept of representations can be used to handle structures of continuum cardinality. In this context the choice of representations is very sensitively related to the underlying notion of approximation, hence to topology. In this paper we summarize some basic ideas of the representation based approach to computable analysis and we introduce an abstract and purely set theoretic characterization of this theory which can be considered as a generalization of the classical concept of µ-recursive functions. Together with this characterization we introduce the notion of a perfect topological structure. In particular, these structures are effectively categorical, i.e. they characterize their own computability theory. Important examples of perfect structures are provided by metric spaces and additional attention is paid to their effective subsets.

Topological properties of real number representations

We prove three results about representations of real numbers (or elements of other topological spaces) by infinite strings. Such representations are useful for the description of real number computations performed by digital computers or by Turing machines. First, we show that the so-called admissible representations, a topologically natural class of representations introduced by Kreitz and Weihrauch, are essentially the continuous extensions (with a well-behaved domain) of continuous and open representations. Second, we show that there is no admissible representation of the real numbers such that every real number has only finitely many names. Third, we show that a rather interesting property of admissible real number representations holds true also for a certain non-admissible representation, namely for the naive Cauchy representation: the property that continuity is equivalent to relative continuity with respect to the representation.

Computable invariance

In Computable Analysis each computable function is continuous and computably invariant , i.e. it maps computable points to computable points. On the other hand, discontinuity is a suucient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discon-tinuous function is suuciently discontinuous and suuciently eeective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf Pour-El & Richards (cf. 20]). We apply our theorem to prove that several set-valued operators are not computably invariant.

Plottable Real Number Functions and the Computable Graph Theorem

The Graph Theorem of classical recursion theory states that a total function on the natural numbers is computable if and only if its graph is recursive. It is known that this result can be generalized to real number functions where it has an important practical interpretation: the total computable real number functions are precisely those which can be effectively plotted with any given resolution. We generalize the Graph Theorem to appropriate partial real number functions and even further to functions defined on certain computable metric spaces. Besides the nonuniform version of the Graph Theorem which logically relates computability properties of the function and computability properties of its graph, we also discuss the uniform version: given a program of a function, can we algorithmically derive a description of its graph? And, vice versa, given a description of the graph, can we derive a program of the function? While the passage from functions to graphs is always computable, the inverse direction from graphs to functions is problematic, and it turns out that the answers to the uniform and the nonuniform questions do not coincide. We prove that in both cases certain topological and computational properties (such as compactness or effective local connectedness) are sufficient for a positive answer, and we provide counterexamples which show that the corresponding properties are not superfluous. Additionally, we briefly discuss the special situation of the linear case.