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A Functional Approach to Computability on Real Numbers
, 1993
"... The aim of this thesis is to contribute to close the gap existing between the theory of computable analysis and actual computation. In order to study computability over real numbers we use several tools peculiar to the theory of programming languages. In particular we introduce a special kind of ty ..."
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The aim of this thesis is to contribute to close the gap existing between the theory of computable analysis and actual computation. In order to study computability over real numbers we use several tools peculiar to the theory of programming languages. In particular we introduce a special kind of typed lambda calculus as an appropriate formalism for describing computations on real numbers. Furthermore we use domain theory, to give semantics to this typed lambda calculus and as a conseguence to give a notion of computability on real numbers. We discuss the adequacy of ScottDomains as domains for representing real numbers. We relate the Scott topology on such domains to the euclidean topology on IR. Domain theory turns out to be useful also in the study of higher order functions. In particular one of the most important results contained in this thesis concerns the characterisation of the topological properties of the computable higher order functions on reals. Our approach allows more...
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation
, 1996
"... ..."
Hardware Speedups in Long Integer Multiplication
 Computer Architecture News
, 1990
"... We present various experiments in Hardware/Software design tradeoffs met in speeding up long integer multiplications. This work spans over a year, with more than 12 different hardware designs tested and measured. To implement these designs, we rely on our PAM (for Programmable Active Memory, see [BR ..."
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We present various experiments in Hardware/Software design tradeoffs met in speeding up long integer multiplications. This work spans over a year, with more than 12 different hardware designs tested and measured. To implement these designs, we rely on our PAM (for Programmable Active Memory, see [BRV]) technology which provides us with a 50 millisecond turnaround time silicon foundry for implementing up to 50K gate logic designs fully equipped with fast local RAM and host bus interface. First, we demonstrate how a simple hardware 512 bits integer multiplier coupled with a low end workstation host yields performance on long arithmetic superior to that of the fastest computers for which we could obtain actual benchmark figures. Second, we specialize this hardware in order to speedup one specific application of long integer arithmetic, namely RivestShamir Adleman publickey cryptography [RSA]. We demonstrate how a single host driving 3 differently configured PAM boards delivers RSA enc...
Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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Cited by 20 (0 self)
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 19 (1 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the
A golden ratio notation for the real numbers
, 1991
"... Several methods to perform exact computations on real numbers have been proposed in the literature. In some of these methods real numbers are represented by infinite (lazy) strings of digits. It is a well known fact that, when this approach is taken, the standard digit notation cannot be used. New f ..."
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Several methods to perform exact computations on real numbers have been proposed in the literature. In some of these methods real numbers are represented by infinite (lazy) strings of digits. It is a well known fact that, when this approach is taken, the standard digit notation cannot be used. New forms of digit notations are necessary. The usual solution to this representation problem consists in adding new digits in the notation, quite often negative digits. In this article we present an alternative solution. It consists in using non natural numbers as “base”, that is, in using a positional digit notation where the ratio between the weight of two consecutive digits is not necessarily a natural number, as in the standard case, but it can be a rational or even an irrational number. We discuss in full detail one particular example of this form of notation: namely the one having two digits (0 and 1) and the golden ratio as base. This choice is motivated by the pleasing properties enjoyed by the golden ratio notation. In particular, the algorithms for the arithmetic operations are quite simple when this notation is used.
Number Computability and Domain Theory
 Information and Computation
, 1996
"... We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. Thi ..."
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We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods. Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scotttopology and the standard topologies on the real line and on the space of continuous functions on reals is stated. An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. 1