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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 4 (3 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
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.
Coinductive Definitions and Real Numbers BSc Final Year Project Report
, 2009
"... Real number computation in modern computers is mostly done via floating point arithmetic which can sometimes produce wildly erroneous results. An alternative approach is to use exact real arithmetic whose results are guaranteed correct to any userspecified precision. It involves potentially infini ..."
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Real number computation in modern computers is mostly done via floating point arithmetic which can sometimes produce wildly erroneous results. An alternative approach is to use exact real arithmetic whose results are guaranteed correct to any userspecified precision. It involves potentially infinite data structures and has therefore in recent years been studied using the mathematical field of universal coalgebra. However, while the coalgebraic definition principle corecursion turned out to be very useful, the coalgebraic proof principle coinduction is not always sufficient in the context of exact real arithmetic. A new approach recently proposed by Berger in [3] therefore combines the more general settheoretic coinduction with coalgebraic corecursion. This project extends Berger’s approach from numbers in the unit interval to the whole real line and thus further explores the combination of coalgebraic corecursion and settheoretic coinduction in the context of exact real arithmetic. We propose a coinductive strategy for studying arithmetic operations on the