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 found...

We describe computational science as an interdisciplinary approach to doing science on computers. Our purpose is to introduce computational science as a legitimate interest of computer scientists. We present a foundation for computational science based on the need to incorporate computation at the scientific level; i.e., computational aspects must be considered when a model is formulated. We next present some obstacles to computer scientists' participation in computational science, including a cultural bias in computer science that inhibits participation. Finally, we look at some areas of conventional computer science and indicate areas of mutual interest between computational science and computer science. Keywords: education, computational science. 1 What is Computational Science ? In December, 1991, the U. S. Congress passed the High Performance Computing and Communications Act, commonly known as the HPCC . This act focuses on several aspects of computing technology, but two have...

We give a detailed treatment of the “bit-model ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis. 1

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ! defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with \Sigma b 1 -induction over PV ! and use this to reestablish that the provably total functions in this system are in polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. 1 Introduction Cook and Urquhart's system PV ! [3] is a simply-typed lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomial-time computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei...

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initial-value problem { dx = f(t, x) dt x(t0) = x0, where E is an open subset of R m+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. 1.

: In the present work the notion of the computable (primitive recursive, polynomially time computable) p--adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p--adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterization of the indices representing the field of computable p--adic numbers is established. Keywords: Computable numbers, computable (primitive recursive, polynomially time computable p--adic numbers, p--adically closed fields, notation systems. 1 Introduction The present work brings together two ideas, namely type two computability, and p--adic fields. We start with a brief review of the notion a computable number and of a p--adic field as a background. The basic idea for computable real numbers is contained in Turing's fundamental paper [20], where he introduced the notion of a computable (or...