Philipp Sünderhauf

A domain-theoretic approach to computability on the real line

In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of Stoltenberg-Hansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turing-machine based approach which dates back to Grzegorczyk and Lacombe, is used by Pour-El & Richards in their foundational work on computable analysis, and, moreover, is the standard notion of computability among physicists as in the work of Penrose. Our framework makes it possible to capture partial functions in an elegant way and it extends to the complex numbers and the n-dimensional Euclidean space.

On the Duality of Compact vs. Open

It is a pleasant fact that Stone‐duality may be described very smoothly when restricted to the category of compact spectral spaces: The Stone‐duals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations.

Computable banach spaces via domain theory

This paper extends the order-theoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded. We show that the domain-theoretic computability theory is equivalent to the well-established approach by Pour-El and Richards.

Lazy computation with exact real numbers

We provide a semantical framework for exact real arithmetic using linear fractional transformations on the extended real line. We present an extension of PCF with a real type which introduces an eventually breadth-first strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if, rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the one-point compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of single-valued elementary functions.

Uniform approximation of topological spaces

Abstract We sharpen the notion of a quasi-uniform space to spaces which carry with them functional means of approximating points, opens and compacts. Assuming nothing but sobriety, the requirement of uniform approximation ensures that such spaces are compact ordered (in the sense of Nachbin). We study uniformly approximated spaces with the means of topology, uniform topology, order theory and locale theory. In each case it turns out that one can give a succinct and meaningful characterization. This leads us to believe that uniform approximation is indeed a concept of central importance.

SMYTH COMPLETENESS IN TERMS OF NETS: THE GENERAL CASE

Abstract Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net strongly converges. As we are dealing with typically non-symmetric spaces, this is not an instance of the classical net-filter translation in general topology.

Tensor Products and Powerspaces in Quantitative Domain Theory

Abstract One approach to quantitative domain theory is the thesis that the underlying boolean logic of ordinary domain theory which assumes only values in the set {true, false} is replaced by a more elaborate logic with values in a suitable structure Ν. (We take Ν to be a value quantale.) So the order ⊑ is replaced by a generalised quasi-metric d, assigning to a pair of points the truth value of the assertion x ⊑ y. In this paper, we carry this thesis over to the construction of powerdomains. This means that we assume the membership relation ∈ to take its values in Ν. This is done by requiring that the value quantale Ν carries the additional structure of a semiring. Powerdomains are then constructed as free modules over this semiring. For the case that the underlying logic is the logic of ordinary domain theory our construction reduces to the familiar Hoare powerdomain. Taking the logic of quasi-metric spaces, i.e. Ν = [0, ∞] with usual addition and multiplication, reveals a close connection to the powerdomain of extended probability measures. As scalar multiplication need not be nonexpansive we develop the theory of moduli of continuity and m-continuous functions. This makes it also possible to consider functions between quantitative domains with different underlying logic. Formal union is an operation which takes pairs as input, so we investigate tensor products and their behavior with respect to the ideal completion.

Sobriety in Terms of Nets

Sobriety is a subtle notion of completeness for topological spaces: A space is sober if it may be reconstructed from the lattice of its open subsets. The usual criterion to check sobriety involves either irreducible closed subsets or completely prime filters of open sets. This paper provides an alternative possibility, thus trying to make sobriety easier to understand. We define the notion of observative net, which, together with an appropriate convergence notion, characterizes sobriety. As the filter approach does not involve just usual (topological) convergence, this is not an instance of the classical net-filter translation in general topology.

Spaces of valuations as quasimetric domains

Abstract We define a natural quasimetric on the set of continuous valuations of a topological space and investigate it in the spirit of quasimetric domain theory. It turns out that the space of valuations of an (ordinary) algebraic domain D is an algebraic quasimetric domain. Moreover, it is precisely the lower powerdomain of D , where D is regarded as a quasimetric domain. The essential tool for proving these results is a generalization of the Splitting Lemma which characterizes the quasimetric for simple valuations and holds for valuations on arbitrary topological spaces.

Preface: Volume 24

Abstract The usual implementation of real numbers in todays computers as floating point numbers has the well-known deficiency that most numbers can only be represented up to some fixed accuracy. Consequently, even the basic arithmetic operations cannot be performed exactly, leading to the ubiquitous round-off errors. This is a serious problem in all disciplines where high accuracy calculations are required. One of the ultimate goals of the theoretical and practical research in this area is to overcome these problems by improving the present implementations and algorithms or providing alternatives. This volume contains the proceedings of the Workshop on Real Number Computation which was held in June 19-20, 1998, in Indianapolis, Indiana, USA. The workshop preceded the IEEE Symposium on Logic in Computer Science (LICS). The meeting aimed to present an introduction to the interdisciplinary area of Real Number Computation. The subject is understood in a broad sense and covers various different fields like Recursion Theory, Interval Analysis, Computer Arithmetic, Semantics of Programming Languages, Number Theory, and Numerical Analysis. The workshop was meant to provide researchers from these different communities an opportunity to meet and exchange ideas. More than 30 participants from eight countries made this workshop a succesful event. Contributions to the proceedings were invited after the workshop from all participants and were sent to referees. Comments and suggestions were forwarded to the authors who thus had a chance to amend their texts. The final versions were accepted by the editors. We would like to take this opportunity to thank all referees for their time and cooperation, and Michael Mislove for his invitation to publish the proceedings in the ENTCS series. 23rd August 1999 Abbas Edalat, David Matula and Philipp Sunderhauf, Guest Editors