Results 1  10
of
12
Novel Approaches to Numerical Software with Result Verification
 NUMERICAL SOFTWARE WITH RESULT VERIFICATION, INTERNATIONAL DAGSTUHL SEMINAR, DAGSTUHL
, 2003
"... Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many real ..."
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Cited by 26 (18 self)
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Traditional design of numerical software with result verification is based on the assumption that we know the algorithm ¦¨§� © ©���� £��������� � that transforms input © ©�� into �� � £��������� � ©���� the output, and we £��������� � know the intervals of possible values of the inputs. Many reallife problems go beyond this paradigm. In some cases, we do not have an algorithm ¦, we only know some relation (constraints) between ©� � and. In other cases, in addition to knowing the intervals, we may know some relations between; we may have some information about the probabilities of different values of © � , and we may know the exact values of some of the inputs (e.g., we may know that © £ ���¨�� �). In this paper, we describe the approaches for solving these reallife problems. In Section 2, we describe interval consistency techniques related to handling constraints; in Section 3, we describe techniques that take probabilistic information into consideration, and in Section 4, we overview techniques for processing exact real numbers.
From coinductive proofs to exact real arithmetic
"... Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresp ..."
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Cited by 8 (6 self)
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Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching nonwellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using prooftheoretic methods for obtaining certified algorithms in exact real arithmetic. 1
Domain theoretic solutions of initial value problems for unbounded vector fields
 Proc. MFPS XXI, Electr. Notes in Theoret. Comp. Sci
, 2005
"... This paper extends the domain theoretic method for solving initial value problems, described in [8], to unbounded vector fields. Based on a sequence of approximations of the vector field, we construct two sequences of piecewise linear functions that converge exponentially fast from above and below t ..."
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Cited by 7 (5 self)
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This paper extends the domain theoretic method for solving initial value problems, described in [8], to unbounded vector fields. Based on a sequence of approximations of the vector field, we construct two sequences of piecewise linear functions that converge exponentially fast from above and below to the classical solution of the initial value problem. We then show, how to construct approximations of the vector field. First, we show, that fast convergence is preserved under composition of approximations, if the approximated functions satisfy an additional property, which we call “Hausdorff Lipschitz from below”. In particular, this frees us from the need to work with maximal extensions of classical functions. In a second step, we show how to construct approximations that satisfy this condition from a given computable vector field. 1
Coinduction for Exact Real Number Computation
, 2007
"... This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the corr ..."
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Cited by 5 (4 self)
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This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio. Key words: Exact real number computation, coinduction, corecursion, signed digit streams. 1
Basic Algorithms of Computational Geometry with Imprecise Input
, 2005
"... The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is c ..."
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Cited by 4 (0 self)
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The domaintheoretic model of computational geometry provides us with continuous and computable predicates and binary operations. It can also be used to generalise the theory of computability for real numbers and real functions into geometric objects and geometric operations. A geometric object is computable if it is the effective limit of a sequence of finitary partial objects of the same type as the original object. We are also provided with two different quantitative measures for approximation using the Hausdorff metric and the Lebesgue measure. In this thesis, we introduce a new data type to capture imprecise data or approximate points on the plane, given in the shape of compact convex polygons. This data type in particular includes rectangular approximation and is invariant under linear transformations of coordinate system. Based on the new data type, we define the notion of a number of partial geometric operations, including partial perpendicular bisector and partial disc and we show that these operations and the convex hull, Delaunay triangulation and Voronoi diagram are Hausdorff and Scott continuous and nestedly Hausdorff and Lebesgue computable. We develop algorithms to obtain the partial convex hull, partial Delaunay triangulation and partial Voronoi diagram. We prove that the complexity of the partial convex hull is N log N in 2D and 3D, whereas the partial Delaunay triangulation and partial Voronoi diagram algorithms for nondegenerate data have the same complexity as their classical counterparts. 2
Constructive Analysis with Witnesses
"... Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completenes ..."
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Cited by 3 (1 self)
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Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5
Proofs, programs, processes
"... Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming la ..."
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Cited by 3 (2 self)
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Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation. 1
Computer Arithmetic Based on Continued Logarithms
 in Proceedings of PAD 2006 (Počítačové Architektury a Diagnostika), Papradno
, 2006
"... Abstract. The paper deals with efficient hardware implementation of exact arithmetic. This kind of arithmetic represents the most convenient approach to scientific computing, but the complexity of underlying algorithms and thus the performance issues limit the set of solvable problems. We demonstrat ..."
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Cited by 1 (0 self)
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Abstract. The paper deals with efficient hardware implementation of exact arithmetic. This kind of arithmetic represents the most convenient approach to scientific computing, but the complexity of underlying algorithms and thus the performance issues limit the set of solvable problems. We demonstrate on a particular example of continued logarithms that direct hardware implementation may help to achieve considerable performance gains. We document this fact by practical results of our research and use them as a starting point for discussion on possible directions of further work.
Realisability and adequacy for (co)induction
"... Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in th ..."
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Cited by 1 (1 self)
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Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further nontrivial applications in computable analysis. 1
Coinductive Proofs for Basic Real Computation Tie Hou
"... Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof ..."
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Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.