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Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
A New Representation for Exact Real Numbers
, 1997
"... We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the rea ..."
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Cited by 42 (8 self)
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We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first socalled sign matrix determines an interval on which the real number lies. The subsequent socalled digit matrices have nonnegative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.
Semantics of Exact Real Arithmetic
, 1997
"... In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the exten ..."
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Cited by 29 (8 self)
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In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.
Lazy Functional Algorithms for Exact Real Functionals
 Lec. Not. Comput. Sci
, 1998
"... . We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a ..."
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Cited by 23 (0 self)
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. We show how functional languages can be used to write programs for realvalued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory. 1 Introduction In exact real number computation, infinite representations of reals are employed to avoid the usual rounding errors that are inherent in floating point computation [46, 17]. For certain real number computations that are highly sensitive to small variations in the input, such rounding errors become inordinately large and the use of floatingpoint algorithms can lead to completely erroneous results [1, 14]. In such situations, exact real number computation provides guaranteed correctness, although at the (probably...
Contractivity of Linear Fractional Transformations
 Third Real Numbers and Computers Conference (RNC3
, 1998
"... One possible approach to exact real arithmetic is to use linear fractional transformations (LFT's) to represent real numbers and computations on real numbers. Recursive expressions built from LFT's are only convergent (i.e., denote a welldefined real number) if the involved LFT's are ..."
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Cited by 8 (3 self)
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One possible approach to exact real arithmetic is to use linear fractional transformations (LFT's) to represent real numbers and computations on real numbers. Recursive expressions built from LFT's are only convergent (i.e., denote a welldefined real number) if the involved LFT's are sufficiently contractive. In this paper, we define a notion of contractivity for LFT's. It is used for convergence theorems and for the analysis and improvement of algorithms for elementary functions. Keywords : Exact Real Arithmetic, Linear Fractional Transformations 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [8, 17, 11, 14, 12, 6]. Onedimensional LFT's x 7! ax+c bx+d are used in the representation of real numbers and to implement basic unary functions, while twodimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as addition and multiplication, and can be combined to obtain infinite expression trees ...
Lazy Computation with Exact Real Numbers
 PROCEEDINGS OF THE THIRD ACM SIGPLAN INTERNATIONAL CONFERENCE ON FUNCTIONAL PROGRAMMING (ICFP98), VOLUME 34, 1 OF ACM SIGPLAN NOTICES
, 1997
"... We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst strategy for lazy evaluation of exact real numbers. In ..."
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Cited by 8 (3 self)
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We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst 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 onepoint 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 singlevalued elementary functions.
The Appearance of Big Integers in Exact Real Arithmetic based on Linear Fractional Transformations
 In Proc. Foundations of Software Science and Computation Structures (FoSSaCS '98), volume 1378 of LNCS
, 1997
"... . 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 nu ..."
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Cited by 7 (4 self)
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. 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. 1 Introduction Linear Fractional Transformations (LFT's) provide an elegant approach to real number arithmetic [8, 16, 11, 14, 12, 6]. Onedimensional LFT's x 7! ax+c bx+d are used as digits and to implement basic functions, while twodimensional LFT's (x; y) 7! axy+cx+ey+g bxy+dx+fy+h provide binary operations such as addition and multiplication, and can be combined to infinite expression trees denoting transcendental functions. In Section 2, we present the details of the LFT approach. This provides the background for understanding the r...
Verified Real Number Calculations: A Library for Interval Arithmetic
, 2007
"... Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally est ..."
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Cited by 6 (1 self)
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Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theoremprover interaction.