Results 1  10
of
17
Abstract versus concrete computation on metric partial algebras
 ACM Transactions on Computational Logic
, 2004
"... Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We pr ..."
Abstract

Cited by 32 (19 self)
 Add to MetaCart
Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations. With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ‘while ’ programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability. The theory of abstract models is stable: there are many models of computation, and
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
(Show Context)
Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
Computability of banach space principles
 INFORMATIK BERICHTE 286, FERNUNIVERSITÄT HAGEN, FACHBEREICH INFORMATIK
, 2001
"... We investigate the computable content of certain theorems which are sometimes called the “principles ” of the theory of Banach spaces. Among these the main theorems are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also study closely related theorems, as ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
We investigate the computable content of certain theorems which are sometimes called the “principles ” of the theory of Banach spaces. Among these the main theorems are the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also study closely related theorems, as Banach’s Inverse Mapping Theorem, the Theorem on Condensation of Singularities and the BanachSteinhaus Theorem. From the computational point of view these theorems are interesting, since their classical proofs rely more or less on the Baire Category Theorem and therefore they count as “nonconstructive”. These theorems have already been studied in Bishop’s constructive analysis but the picture that we can draw in computable analysis differs in several points. On the one hand, computable analysis is based on classical logic and thus can apply stronger principles to prove that certain operations are computable. On the other hand, classical logic enables
Domain Representations of Partial Functions, with Applications to Spatial Objects and Constructive Volume Geometry
, 2000
"... A partial spatial object is a partial map from space to data. Data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry (CVG), where space and data are subspaces o ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
A partial spatial object is a partial map from space to data. Data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry (CVG), where space and data are subspaces of # dimensional Euclidean space. We investigate the computability of partial spatial object data types, in general and in volume graphics, using the theory of effective domain representations for topological algebras. The basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces. We prove theorems about partial functions on regular Hausdorff spaces and their domain representations, and apply the results to partial spatial objects and CVG algebras.
Representations versus Numberings: On the Relationship of Two Computability Notions
 THEORETICAL COMPUTER SCIENCE
, 2001
"... This paper gives an answer to Weihrauch's question [31] whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in g ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
This paper gives an answer to Weihrauch's question [31] whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in general. A condition is introduced and for countably based topological T 0 spaces it is shown that exactly the (partial) effective maps meeting the requirement are extendable. For total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directedcomplete partial orders, in which the extendability is already known. In the first case a similar result holds also for partial effective maps, but not in the second.
Computable total functions on metric algebras, universal algebraic specifications and dynamical systems
 THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
, 2005
"... ..."
Effectivity of regular spaces
 Computability and Complexity in Analysis, volume 2064 of Lecture Notes in Computer Science
, 2001
"... Abstract. General methods of investigating effectivity on regular Hausdorff (T3) spaces is considered. It is shown that there exists a functor from a category of T3 spaces into a category of domain representations. Using this functor one may look at the subcategory of effective domain representation ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. General methods of investigating effectivity on regular Hausdorff (T3) spaces is considered. It is shown that there exists a functor from a category of T3 spaces into a category of domain representations. Using this functor one may look at the subcategory of effective domain representations to get an effectivity theory for T3 spaces. However, this approach seems to be beset by some problems. Instead, a new approach to introducing effectivity to T3 spaces is given. The construction uses effective retractions on effective Scott–Ershov domains. The benefit of the approach is that the numbering of the basis and the numbering of the elements are derived at once. 1
On some problems in computational topology
 Schriften zur Theoretischen Informatik Bericht Nr.0503, Fachberich Mathematik, Universitaet
, 2003
"... Computations in spaces like the real numbers are not done on the points of the space itself but on some representation. If one considers only computable points, i.e., points that can be approximated in a computable way, finite objects as the natural numbers can be used for this. In the case of the r ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Computations in spaces like the real numbers are not done on the points of the space itself but on some representation. If one considers only computable points, i.e., points that can be approximated in a computable way, finite objects as the natural numbers can be used for this. In the case of the real numbers such an indexing can e.g. be obtained by taking the Gödel numbers of those total computable functions that enumerate a fast Cauchy sequence of rational numbers. Obviously, the numbering is only a partial map. It will be seen that this is not a consequence of a bad choice, but is so by necessity. The paper will discuss some consequences. All is done in a rather general topological framework. 1
Computable versions of the uniform boundedness theorem
 Logic Colloquium 2002
, 2006
"... Abstract. We investigate the computable content of the Uniform Boundedness Theorem and of the closely related BanachSteinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequ ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We investigate the computable content of the Uniform Boundedness Theorem and of the closely related BanachSteinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the sequence is “given”. If it is just given with respect to the compact open topology (i.e. if just a sequence of “programs ” is given), then we cannot even compute an upper bound of the uniform bound in general. If, however, the pointwise bounds are available as additional input information, then we can effectively compute an upper bound of the uniform bound. Additionally, we prove an effective version of the contraposition of the Uniform Boundedness Theorem: given a sequence of linear bounded operators which is not uniformly bounded, we can effectively find a witness for the fact that the sequence is not pointwise bounded. As an easy application of this theorem we obtain a computable function whose Fourier series does not converge. §1. Introduction. In this paper we want to study the computational content of some theorems of functional analysis. The Uniform Boundedness Theorem is one of the central theorems of functional analysis and it has first been published in Banach’s thesis [1].
Computable Real Functions: Type 1 Computability Versus Type 2 Computability
, 1996
"... Based on the Turing machine model there are essentially two different notions of computable functions over the real numbers. The effective functions are defined only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The effectively conti ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Based on the Turing machine model there are essentially two different notions of computable functions over the real numbers. The effective functions are defined only on computable real numbers and are Type 1 computable with respect to a numbering of the computable real numbers. The effectively continuous functions may be defined on arbitrary real nunbers. They are exactly those functions which are Type 2 computable with respect to an appropriate representation of the real numbers. We characterize the Type 2 computable functions on computable real numbers as exactly those Type 1 computable functions which satisfy a certain additional condition concerning their domain of definition. Our result is a sharp strengthening of the wellknown continuity result of Tseitin and Moschovakis for effective functions. The result is presented for arbitrary computable metric spaces. 1 Introduction In this paper we compare two approaches for defining computability of functions between computable metric ...