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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 ..."
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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.
A LargeScale Experiment in Executing Extracted Programs
"... It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program ..."
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Cited by 9 (2 self)
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It is a wellknown fact that algorithms are often hidden inside mathematical proofs. If these proofs are formalized inside a proof assistant, then a mechanism called extraction can generate the corresponding programs automatically. Previous work has focused on the difficulties in obtaining a program from a formalization of the Fundamental Theorem of Algebra inside the Coq proof assistant. In theory, this program allows one to compute approximations of roots of polynomials. However, as we show in this work, there is currently a big gap between theory and practice. We study the complexity of the extracted program and analyze the reasons of its inefficiency, showing that this is a direct consequence of the approach used throughout the formalization.
Programming and certifying the CAD algorithm inside the coq system
 Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl Seminar Proceedings, Schloss Dagstuhl
, 2005
"... Abstract. A. Tarski has shown in 1975 that one can perform quantifier elimination in the theory of real closed fields. The introduction of the Cylindrical Algebraic Decomposition (CAD) method has later allowed to design rather feasible algorithms. Our aim is to program a reflectional decision proced ..."
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Abstract. A. Tarski has shown in 1975 that one can perform quantifier elimination in the theory of real closed fields. The introduction of the Cylindrical Algebraic Decomposition (CAD) method has later allowed to design rather feasible algorithms. Our aim is to program a reflectional decision procedure for the Coq system, using the CAD, to decide whether a (possibly multivariate) system of polynomial inequalities with rational coefficients has a solution or not. We have therefore implemented various computer algebra tools like gcd computations, subresultant polynomial or Bernstein polynomials.
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 ..."
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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.
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"... This paper develops machinery necessary to mechanically import arbitrary functional programs into Coq’s type theory, manually strengthen their specifications with additional proofs, and then mechanicaly reextract the newlycertified program in a form which is as efficient as the original program. I ..."
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This paper develops machinery necessary to mechanically import arbitrary functional programs into Coq’s type theory, manually strengthen their specifications with additional proofs, and then mechanicaly reextract the newlycertified program in a form which is as efficient as the original program. In order to facilitate this goal, the coinductive technique of [Cap05] is modified to form a monad whose operators are the constructors of a coinductive type rather than functions defined over the type. The inductive invariant technique of [KM03] is extended to allow optional “after the fact ” termination proofs. These proofs inhabit members of Prop, and therefore do not affect extracted code. Compared to [Cap05], the new monad makes it possible to directly represent unrestricted recursion without violating productivity requirements [Gim95], and it produces efficient code via Coq’s extraction mechanism. The disadvantages of this technique include reliance on the JMeq axiom [McB00] and a significantly more complex notion of equality. The resulting technique is packaged as a Coq library, and is suitable for formalizing programs written in any sideeffectfree functional language with callbyvalue semantics.
A LargeScale Experiment in Executing Extracted Programs
"... 1 Introduction Several approaches can be used for certifying software. A first one, perhaps the most natural, is to start with an handwritten program and then inspect it formally in a suitable logical system, like Hoare logic. But there exists an alternative approach where one needs not write the pr ..."
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1 Introduction Several approaches can be used for certifying software. A first one, perhaps the most natural, is to start with an handwritten program and then inspect it formally in a suitable logical system, like Hoare logic. But there exists an alternative approach where one needs not write the program, but rather obtains it automatically from a mathematical proof. This automatic transformation of proofs into correctbyconstruction programs is called (program) extraction.
CO620
"... When a number is represented as a continued fraction, then it comes with a natural error bound. Continued fractions can be expressed as digit streams. Arbitrary precision can be achieved by truncating the stream appropriately. Introducing more terms will refine the representation whilst preserving t ..."
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When a number is represented as a continued fraction, then it comes with a natural error bound. Continued fractions can be expressed as digit streams. Arbitrary precision can be achieved by truncating the stream appropriately. Introducing more terms will refine the representation whilst preserving the ability for further refinement. The value of continued fraction arithmetic has been recognized by the functional programming community, because continued fractions can be naturally implemented as lazy streams, but is not as widely known in logic programming. Delay declarations can be used to orchestrate the control needed to compute numeric results lazily to the required degree of precision. Irrational numbers can be represented by infinite continued fractions, which, if they have recurring patterns, can be represented exactly by rational trees. This project demonstrates how continued fraction arithmetic works and how it can be implemented using logic programming features to achieve the desired precision of a result. 1.
Admissible Digit Sets and a Modified Stern–Brocot Representation
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set ” yields an admissible representation of [0, +∞]. 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 Stern–Brocot tree. We show how we can modify the usual Stern–Brocot representation to yield a ternary admissible digit set.