policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.

We introduce, in Part I, a number representation suitable for exact real number computation, consisting of an exponent and a mantissa, which is an in nite stream of signed digits, based on the interval [ 1; 1]. Numerical operations are implemented in terms of linear fractional transformations (LFT's). We derive lower and upper bounds for the number of argument digits that are needed to obtain a desired number of result digits of a computation, which imply that the complexity of LFT application is that of multiplying n-bit integers. In Part II, we present an accessible account of a domain-theoretic approach to computational geometry and solid modelling which provides a data-type for designing robust geometric algorithms, illustrated here by the convex hull algorithm.

Abstract not found

We present a new model of geometric computation which supports the design of robust algorithms for exact real number input as well as for input with uncertainty, i.e. partial input. In this framework, we show that the convex hull of N computable real points in R^d is indeed computable. We provide a robust algorithm which, given any set of N partial inputs, i.e. N dyadic or rational rectangles, approximating these points, computes the partial convex hull in time O(N log N) in 2d and 3d. As the rectangles are refined to the N points, the sequence of partial convex hulls converges effectively both in the Hausdorff metric and the Lebesgue measure to the convex hull of the N points.

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.

Abstract. We introduce a domain-theoretic computational model for multivariable differential calculus, which for the first time gives rise to data types for differentiable functions. The model, a continuous Scott domain for differentiable functions of n variables, is built as a sub-domain of the product of n + 1 copies of the function space on the domain of intervals by tupling together consistent information about locally Lipschitz (piecewise differentiable) functions and their differential properties (partial derivatives). The main result of the paper is to show, in two stages, that consistency is decidable on basis elements, which implies that the domain can be given an effective structure. First, a domain-theoretic notion of line integral is used to extend Green’s theorem to interval-valued vector fields and show that integrability of the derivative information is decidable. Then, we use techniques from the theory of minimal surfaces to construct the least and the greatest piecewise linear functions that can be obtained from a tuple of n + 1 rational step functions, assuming the integrability of the n-tuple of the derivative part. This provides an algorithm to check consistency on the rational basis elements of the domain, giving an effective framework for multi-variable differential calculus. 1

Abstract not found

Key words Computability, recursive analysis, regular set, set operators.

C1 norm to the inverse func-tion. A similar result holds for implicit functions. Combined with the domain-theoretic model for computationalgeometry, this provides a robust technique for construction of curves and surfaces in geometric modelling and CAD. 1.

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.