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A DomainTheoretic Approach to Computability on the Real Line
, 1997
"... 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 ..."
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Cited by 43 (8 self)
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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 StoltenbergHansen 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 Turingmachine based approach which dates back to Grzegorczyk and Lacombe, is used by PourEl & Richards in their found...
Computing over the reals: Foundations for scientific computing
 Notices of the AMS
"... We give a detailed treatment of the “bitmodel ” 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 BlumShubSmale model. In the final section we d ..."
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Cited by 32 (3 self)
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We give a detailed treatment of the “bitmodel ” 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 BlumShubSmale model. In the final section we discuss the issue of whether physical systems could defeat the ChurchTuring Thesis. 1
Science, Computational Science and Computer Science: At a Crossroads
 Comm. ACM
, 1993
"... 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 s ..."
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Cited by 26 (2 self)
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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...
Computability, noncomputability and undecidability of maximal intervals of IVPs
 Trans. Amer. Math. Soc
"... Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue 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 o ..."
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Cited by 15 (14 self)
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Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initialvalue 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 initialvalue 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.
An Application of CategoryTheoretic Semantics to the Characterisation of Complexity Classes Using HigherOrder Function Algebras
, 1997
"... We use the category of presheaves over PTIMEfunctions in order to show that Cook and Urquhart's higherorder function algebra PV ! defines exactly the PTIMEfunctions. As a byproduct we obtain a syntaxfree generalisation of PTIMEcomputability to higher types. By restricting to sheaves for a sui ..."
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Cited by 11 (6 self)
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We use the category of presheaves over PTIMEfunctions in order to show that Cook and Urquhart's higherorder function algebra PV ! defines exactly the PTIMEfunctions. As a byproduct we obtain a syntaxfree generalisation of PTIMEcomputability 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 categorytheoretic approach to a new higherorder extension of BellantoniCook's system BC of safe recursion. 1 Introduction Cook and Urquhart's system PV ! [3] is a simplytyped lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomialtime computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... 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 ..."
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Cited by 11 (5 self)
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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.
COMPUTATIONAL UNSOLVABILITY OF DOMAINS OF ATTRACTION OF NONLINEAR SYSTEMS
, 2009
"... Let S be the domain of attraction of a computable and asymptotically stable hyperbolic equilibrium point of the nonlinear system ˙x = f(x). We show that the problem of determining S is computationally unsolvable. We also present an upper bound of the degree of unsolvability of this problem. ..."
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Cited by 6 (6 self)
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Let S be the domain of attraction of a computable and asymptotically stable hyperbolic equilibrium point of the nonlinear system ˙x = f(x). We show that the problem of determining S is computationally unsolvable. We also present an upper bound of the degree of unsolvability of this problem.
Computable padic Numbers
, 1999
"... : In the present work the notion of the computable (primitive recursive, polynomially time computable) padic 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 padically c ..."
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: In the present work the notion of the computable (primitive recursive, polynomially time computable) padic 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 padically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterization of the indices representing the field of computable padic numbers is established. Keywords: Computable numbers, computable (primitive recursive, polynomially time computable padic numbers, padically closed fields, notation systems. 1 Introduction The present work brings together two ideas, namely type two computability, and padic fields. We start with a brief review of the notion a computable number and of a padic 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...