Exact real arithmetic: a case study in higher order programming

Boehm, Hans-J.; Cartwright, Robert; Riggle, Mark; O’Donnell, Michael

doi:10.1145/319838.319860

Cited by 73 publications

(42 citation statements)

References 14 publications

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“…There are many ways to represent real numbers as infinite objects [3,2,4,5]. Here, we are only concerned with representations as infinite streams of "digits".…”

Section: From Digit Streams To Linear Fractional Transformationsmentioning

confidence: 99%

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The appearance of big integers in exact real arithmetic based on Linear Fractional Transformations

Heckmann

1998

Lecture Notes in Computer Science

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Abstract. One possible approach to exact real arithmetic is to use linear fractional transformations to represent real numbers and computations on real numbers. In this paper, we show that the bit sizes of the (integer) parameters of nearly all transformations used in computations are proportional to the number of basic computational steps executed so far. Here, a basic step means consuming one digit of the argument(s) or producing one digit of the result.

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“…There are many ways to represent real numbers as infinite objects [3,2,4,5]. Here, we are only concerned with representations as infinite streams of "digits".…”

Section: From Digit Streams To Linear Fractional Transformationsmentioning

confidence: 99%

“…There are several different stream representations which can be grouped into two large families: variations of the familiar decimal representation [1,3,2,5,7,11,10], and continued fraction expansions [8,16,9].…”

Section: From Digit Streams To Linear Fractional Transformationsmentioning

confidence: 99%

The appearance of big integers in exact real arithmetic based on Linear Fractional Transformations

Heckmann

1998

Lecture Notes in Computer Science

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“…Rutten's study [5] of power series from an automatatheoretic perspective explored the expressive power of automata representing analytic objects. Also, the idea of representing exact reals by streams has been pursued in depth [6], [7]. In contrast, expressiveness is not the real concern of this work.…”

Section: Introductionmentioning

confidence: 99%

Regular Real Analysis

Chaudhuri

Sankaranarayanan

Vardi

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-We initiate the study of regular real analysis, or the analysis of real functions that can be encoded by automata on infinite words. It is known that ω-automata can be used to represent relations between real vectors, reals being represented in exact precision as infinite streams. The regular functions studied here constitute the functional subset of such relations.We show that some classic questions in function analysis can become elegantly computable in the context of regular real analysis. Specifically, we present an automatatheoretic technique for reasoning about limit behaviors of regular functions, and obtain, using this method, a decision procedure to verify the continuity of a regular function. Several other decision procedures for regular functions-for finding roots, fixpoints, minima, etc.-are also presented. At the same time, we show that the class of regular functions is quite rich, and includes functions that are highly challenging to encode using traditional symbolic notation.

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“…Although the theory of computable analysis can be considered a well-developed subject, there have been so far very few attempts of implementing computable analysis on digital computers, see Boehm and Cartwright, Grue, Vuillemin, [BCRO86], [Boe87], [Gru88], [Vui88]. Such implementations should lead to the realization of "exact real number computation".…”

Section: Introductionmentioning

confidence: 99%

Real number computability and domain theory

Gianantonio

1993

Lecture Notes in Computer Science

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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 Scott-topology 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.

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Exact real arithmetic: a case study in higher order programming

Cited by 73 publications

References 14 publications

The appearance of big integers in exact real arithmetic based on Linear Fractional Transformations

The appearance of big integers in exact real arithmetic based on Linear Fractional Transformations

Regular Real Analysis

Real number computability and domain theory

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