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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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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.
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.