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Lazy computation with exact real numbers

Authors:

Peter John Potts,

Philipp SünderhaufAuthors Info & Claims

ACM SIGPLAN Notices, Volume 34, Issue 1

Pages 185 - 194

https://doi.org/10.1145/291251.289441

Published: 29 September 1998 Publication History

Abstract

We provide a semantical framework for exact real arithmetic using linear fractional transformations on the extended real line. We present an extension of PCF with a real type which introduces an eventually breadth-first strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if, rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the one-point compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of single-valued elementary functions.

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Edalat A(2021)Smooth Approximation of Lipschitz Maps and Their SubgradientsJournal of the ACM10.1145/348180569:1(1-32)Online publication date: 22-Dec-2021
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Geuvers HNiqui MSpitters BWiedijk F(2007)Constructive analysis, types and exact real numbersMathematical Structures in Computer Science10.1017/S096012950600583417:1(3-36)Online publication date: 1-Feb-2007
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Ménissier-Morain V(2005)Arbitrary precision real arithmetic: design and algorithmsThe Journal of Logic and Algebraic Programming10.1016/j.jlap.2004.07.00364:1(13-39)Online publication date: Jul-2005
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Lazy computation with exact real numbers

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Publication History

Published: 29 September 1998

Published in SIGPLAN Volume 34, Issue 1

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Cited By

Edalat A(2021)Smooth Approximation of Lipschitz Maps and Their SubgradientsJournal of the ACM10.1145/348180569:1(1-32)Online publication date: 22-Dec-2021
https://dl.acm.org/doi/10.1145/3481805
Geuvers HNiqui MSpitters BWiedijk F(2007)Constructive analysis, types and exact real numbersMathematical Structures in Computer Science10.1017/S096012950600583417:1(3-36)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.1017/S0960129506005834
Ménissier-Morain V(2005)Arbitrary precision real arithmetic: design and algorithmsThe Journal of Logic and Algebraic Programming10.1016/j.jlap.2004.07.00364:1(13-39)Online publication date: Jul-2005
https://doi.org/10.1016/j.jlap.2004.07.003
Edalat AHeckmann R(2002)Computing with Real NumbersApplied Semantics10.1007/3-540-45699-6_5(193-267)Online publication date: 20-Sep-2002
https://doi.org/10.1007/3-540-45699-6_5
Edalat AMaleki M(2018)Differential Calculus with Imprecise Input and Its Logical FrameworkFoundations of Software Science and Computation Structures10.1007/978-3-319-89366-2_25(459-475)Online publication date: 14-Apr-2018
https://doi.org/10.1007/978-3-319-89366-2_25
Geuvers HNiqui MSpitters BWiedijk F(2007)Constructive analysis, types and exact real numbersMathematical Structures in Computer Science10.1017/S096012950600583417:1(3-36)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.1017/S0960129506005834
Marcial-Romero JEscardó M(2007)Semantics of a sequential language for exact real-number computationTheoretical Computer Science10.1016/j.tcs.2007.01.021379:1-2(120-141)Online publication date: 1-Jul-2007
https://dl.acm.org/doi/10.1016/j.tcs.2007.01.021
Hughes JNiqui M(2006)Admissible digit setsTheoretical Computer Science10.1016/j.tcs.2005.09.059351:1(61-73)Online publication date: 14-Feb-2006
https://dl.acm.org/doi/10.1016/j.tcs.2005.09.059
Niqui M(2005)Formalising exact arithmetic in type theoryProceedings of the First international conference on Computability in Europe: new Computational Paradigms10.1007/11494645_46(368-377)Online publication date: 8-Jun-2005
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Marcial-Romero JEscardo M(2004)Semantics of a sequential language for exact real-number computationProceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.10.1109/LICS.2004.1319637(426-435)Online publication date: 2004
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