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Program extraction in exact real arithmetic

KENJI MIYAMOTO, HELMUT SCHWICHTENBERG
2014 Mathematical Structures in Computer Science  
They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works  ...  Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals.  ...  Conclusion We presented a formal proof of the existence of the average of two real numbers in [−1, 1], as a case study in constructive exact real arithmetic.  ... 
doi:10.1017/s0960129513000327 fatcat:fq2pz63ahzhbph72o65axpvecq

From coinductive proofs to exact real arithmetic: theory and applications

Ulrich Berger, Reinhard Kahle
2011 Logical Methods in Computer Science  
The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers.  ...  Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs.  ...  FROM COINDUCTIVE PROOFS TO EXACT REAL ARITHMETIC: THEORY AND APPLICATIONS 5 In the first example object variables are of sort R while in the second example they are of sort D (hence R and D are the "universal  ... 
doi:10.2168/lmcs-7(1:8)2011 fatcat:xvobbwkxgfcwfjlqixqqhgmzrq

Constructive analysis, types and exact real numbers

HERMAN GEUVERS, MILAD NIQUI, BAS SPITTERS, FREEK WIEDIJK
2007 Mathematical Structures in Computer Science  
We will present some of the problems and solutions of exact real arithmetic varying from concrete implementations, representation and algorithms to various models for real computation.  ...  We then put these models in a uniform framework using realisability, opening the door for the use of type theoretic and coalgebraic constructions both in computing and reasoning about these computations  ...  Program Extraction Correctness is an important issue in the implementation of computable analysis (which in practice currently mostly amounts to the implementation of exact real arithmetic.)  ... 
doi:10.1017/s0960129506005834 fatcat:kdehtcealnbszcfuszmcbzpdhu

Error-free computer solution of certain systems of linear equations

Stephen F. Chang, William J. Kennedy
1987 Journal of Computational and Applied Mathematics  
Then it extracts the exact solution x from x* if the error in the approximation x * is sufficiently small.  ...  In this article we propose a procedure which generates the exact solution for the system Ax = b, where A is an integral nonsingular matrix and b is an integral vector, by improving the initial floating-point  ...  Numerical examples The following three examples demonstrate the extraction procedure. The program is coded in FORTRAN using double precision arithmetic and was run on a CDC Cyber 730 System. e!  ... 
doi:10.1016/0377-0427(87)90002-1 fatcat:c2236bfa75dtzfayf2ltkobuku

The world's shortest correct exact real arithmetic program?

David R. Lester
2012 Information and Computation  
In this paper we present what is thought to be the world's shortest correct exact real arithmetic program.  ...  In addition the program presented here allows beginners to the field to easily experiment with a practical implementation in order to understand some of the issues involved.  ...  Introduction Various attempts have been made to validate exact arithmetics before. Ménissier provides a hand proof of an exact arithmetic [9] .  ... 
doi:10.1016/j.ic.2011.09.004 fatcat:g66trrdoqfgkdjqftcsw4v2uyy

Subject Index

2007 Journal of Discrete Algorithms  
Itai's randomized broadcasting algorithm, 323 Railway optimization Track assignment, 250 Rational Exact arithmetic on the Stern-Brocot tree, 356 Real roots Real roots of univariate polynomials  ...  extracting motifs from biological weighted se- quences, 229 Multilinear Exact arithmetic on the Stern-Brocot tree, 356 Multiple patterns Searching for a set of correlated patterns, 149 Natural language  ... 
doi:10.1016/s1570-8667(07)00076-7 fatcat:wfqxglrznfb6do3wyittd5pfbi

Speeding up Exact Real Arithmetic on Fast Binary Cauchy Sequences by Using Memoization Based on Quantized Precision

Hideyuki Kawabata
2017 Journal of Information Processing  
Exact Real Arithmetic on Fast Binary Cauchy Sequences (FBCSs) provides us a simple and fairly fast way to obtain numerical results of arbitrary precision.  ...  without sacrificing the properties of the arithmetic to be exact arithmetic.  ...  Exact real arithmetic offers a simple way of programming to obtain arbitrarily accurate answers for computable real numbers [7] .  ... 
doi:10.2197/ipsjjip.25.494 fatcat:vxzbmxa3qzaspmxhhdi2m66kyq

On using floating-point computations to help an exact linear arithmetic decision procedure [article]

David Monniaux
2009 arXiv   pre-print
We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the reals or rationals.  ...  State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and program analysis (sparse inequalities, small coefficients  ...  to perform computations in exact arithmetic.  ... 
arXiv:0904.3525v1 fatcat:vgmwnfhd7ngz7ey43tmi5a6qra

On Using Floating-Point Computations to Help an Exact Linear Arithmetic Decision Procedure [chapter]

David Monniaux
2009 Lecture Notes in Computer Science  
We consider the decision problem for quantifier-free formulas whose atoms are linear inequalities interpreted over the reals or rationals.  ...  State-of-the-art SMT solvers use simplex implementations over rational numbers, which perform well for typical problems arising from model-checking and program analysis (sparse inequalities, small coefficients  ...  to perform computations in exact arithmetic.  ... 
doi:10.1007/978-3-642-02658-4_42 fatcat:f5um44i5qja4hpwkwhpqjwk2ge

Implementing Real Numbers With RZ

Andrej Bauer, Iztok Kavkler
2008 Electronical Notes in Theoretical Computer Science  
of exact real arithmetic.  ...  RZ is a tool which translates axiomatizations of mathematical structures to program specifications using the realizability interpretation of logic.  ...  As mentioned earlier, efficient implementations of exact real arithmetic compute in stages.  ... 
doi:10.1016/j.entcs.2008.03.027 fatcat:jahsbjrqdvanrfgjudjgmpkzpi

From Coinductive Proofs to Exact Real Arithmetic [chapter]

Ulrich Berger
2009 Lecture Notes in Computer Science  
This is a pilot study in using proof-theoretic methods for obtaining certified algorithms in exact real arithmetic.  ...  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  ...  Note also that in the proofs we used arithmetic operations on the reals and their arithmetic laws without implementing or proving them.  ... 
doi:10.1007/978-3-642-04027-6_12 fatcat:yixkpnppfbef5mblyenhh54ysa

A coinductive approach to verified exact real number computation

Ulrich Berger, Sion Lloyd
2009 Electronic Communications of the EASST  
We present an approach to verified programs for exact real number computation that is based on inductive and coinductive definitions and program extraction from proofs.  ...  We informally discuss the theoretical background of this method and give examples of extracted programs implementing the translation between the representation by fast converging rational Cauchy sequences  ...  The generation of provably correct programs in exact real number computation is the focus of this paper.  ... 
doi:10.14279/tuj.eceasst.23.331 dblp:journals/eceasst/BergerL09 fatcat:yorfrgpl35gv5btt3bbcg4rfyy

Symbolic execution of floating-point computations

Bernard Botella, Arnaud Gotlieb, Claude Michel
2006 Software testing, verification & reliability  
In this paper, we address the peculiarities of the symbolic execution of program with oatingpoint numbers.  ...  Issues in the symbolic execution of this kind of programs are carefully examined and a constraint solver is described that supports constraints over oating-point numbers.  ...  Gotlieb et al. 25] applied Constraint Logic Programming over nite domains to solve constraints extracted from imperative programs in the tool INKA 26].  ... 
doi:10.1002/stvr.333 fatcat:syrgj5noznhppcrb4swggguf2i

A generic lazy evaluation scheme for exact geometric computations

Sylvain Pion, Andreas Fabri
2011 Science of Computer Programming  
It also relies on multi-precision arithmetic as well as interval arithmetic. We apply our approach to the whole geometry kernel of Cgal.  ...  Our approach is generic and extensible in the sense that it is possible to make it a library that users can apply to their own geometric objects and primitives.  ...  Indeed, the crucial exactness assumption is not satisfied when floating-point arithmetic is used to implement operations on real numbers.  ... 
doi:10.1016/j.scico.2010.09.003 fatcat:nf7ndiyxonaard2ajoxzqzfmpa

Proofs, Programs, Processes [chapter]

Ulrich Berger, Monika Seisenberger
2010 Lecture Notes in Computer Science  
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.  ...  Typical applications of such processes are computations in exact real arithmetic.  ... 
doi:10.1007/978-3-642-13962-8_5 fatcat:rcvrnid7evgwdcntvdbblrcz64
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