Results 1  10
of
18
QArith: Coq formalisation of lazy rational arithmetic
 Types for Proofs and Programs, volume 3085 of LNCS
, 2003
"... Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation use ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation uses advanced machinery of the Coq theorem prover and applies recent developments in formalising general recursive functions. This formalisation highlights the rôle of type theory both as a tool to verify handwritten programs and as a tool to generate verified programs. 1
Exact Arithmetic on the SternBrocot Tree
 NIJMEEGS INSTITUUT VOOR INFORMATICA EN INFORMATEIKUNDE, 2003. HTTP://WWW.CS.RU.NL/RESEARCH/REPORTS/FULL/NIIIR0325.PDF
, 2003
"... In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
In this paper we present the Stern{Brocot tree as a basis for performing exact arithmetic on rational and real numbers. We introduce the tree and mention its relation with continued fractions. Based on the tree we present a binary representation of rational numbers and investigate various algorithms to perform exact rational arithmetic using a simpli ed version of the homographic and the quadratic algorithms [19, 12]. We show generalisations of homographic and quadratic algorithms to multilinear forms in n variables and we prove the correctness of the algorithms. Finally we modify the tree to get a redundant representation for real numbers.
Streaming RepresentationChangers
 LNCS
, 2004
"... Unfolds generate data structures, and folds consume them. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Unfolds generate data structures, and folds consume them.
Incremental Addition in Exact Real Arithmetic
, 1998
"... Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of these approaches. A key property distinguishing the approaches is incrementality: If the accuracy of the result has to be increased in the function approach, computation starts from scratch and all previous calculations have to be disregarded. In contrast, the signed digit approach is incremental, i.e. the previous result is reused and some further digits are computed to increase precision. In this paper, we show how the function approach can be modified, resulting in a hybrid representation where signed digit expansions can be read as functions and vice versa. We develop an algorithm for addition in this setting combining advantages of both approaches. Keywords: Exact real arithmetic, in...
Semantics of QueryDriven Communication of Exact Values1
"... Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducin ..."
Abstract
 Add to MetaCart
Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducing explicit queries. We formalise this approach using protocols of a queryanswer nature. Such protocols enable processes to provide valid approximations with certain accuracy and focusing on certain locality as demanded by the receiving processes through queries. A latticetheoretic denotational semantics of channel and process behaviour is developed. The query space is modelled as a continuous lattice in which the top element denotes the query demanding all the information, whereas other elements denote queries demanding partial and/or local information. Answers are interpreted as elements of lattices constructed over suitable domains of approximations to the exact objects. An unanswered query is treated as an error and denoted using the top element. The major novel characteristic of our semantic model is that it reflects the dependency of answers on queries. This enables the definition and analysis of an appropriate concept of convergence rate, by assigning an effort indicator to each query and a measure of information content to each answer. Thus we capture not only what function a process computes, but also how a process transforms the convergence rates from its inputs to its outputs. In future work these indicators can be used to capture further computational complexity measures. A robust prototype implementation of our model is available.
Compositional Semantics of Dataflow Networks with QueryDriven Communication of Exact Values 1
"... Abstract: We develop and study the concept of dataflow process networks as used for example by Kahn to suit exact computation over data types related to real numbers, such as continuous functions and geometrical solids. Furthermore, we consider communicating these exact objects among processes using ..."
Abstract
 Add to MetaCart
Abstract: We develop and study the concept of dataflow process networks as used for example by Kahn to suit exact computation over data types related to real numbers, such as continuous functions and geometrical solids. Furthermore, we consider communicating these exact objects among processes using protocols of a queryanswer nature as introduced in our earlier work. This enables processes to provide valid approximations with certain accuracy and focusing on certain locality as demanded by the receiving processes through queries. We define domaintheoretical denotational semantics of our networks in two ways: (1) directly, i. e. by viewing the whole network as a composite process and applying the process semantics introduced in our earlier work; and (2) compositionally, i. e. by a fixedpoint construction similar to that used by Kahn from the denotational semantics of individual processes in the network. The direct semantics closely corresponds to the operational semantics of the network (i. e. it is correct) but very difficult to study for concrete networks. The compositional semantics enables compositional analysis of concrete networks, assuming it is correct. We prove that the compositional semantics is a safe approximation of the direct semantics. We also provide a method that can be used in many cases to establish that the two semantics fully coincide, i. e. safety is not achieved through inactivity or meaningless answers. The results are extended to cover recursivelydefined infinite networks as well as nested finite networks. A robust prototype implementation of our model is available.
Exact Real Number Computation Using Linear Fractional Transformations
"... which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy. ..."
Abstract
 Add to MetaCart
(Show Context)
which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.
Admissible Digit Sets and a Modified SternBrocot Representation
, 2004
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "dig ..."
Abstract
 Add to MetaCart
We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such "admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the SternBrocot tree. We show how we can modify the usual SternBrocot representation to yield a ternary admissible digit set.
Wheels
, 2001
"... Filoso e licentiatavhandling We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is in a sense possible. The resulting structure is what is called a wheel. Wheels are similar to rings, but 0x = 0 does not hold in general; the subset fx j 0x = 0g of ..."
Abstract
 Add to MetaCart
Filoso e licentiatavhandling We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is in a sense possible. The resulting structure is what is called a wheel. Wheels are similar to rings, but 0x = 0 does not hold in general; the subset fx j 0x = 0g of any wheel is a commutative ring (or semiring) and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings. Contents 1