Abstract
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first author it was shown how to extract certified algorithms working with the signed digit representations from constructive proofs. In this paper we lay the foundation for doing a similar thing with nonempty compact sets. It turns out that a representation by streams of finitely many digits is impossible and instead trees are needed.
Highlights
Digital representations of real numbers have been widely studied in the literature
[12] Berger and Hou [4] and many others, where a real number in [−1, 1] is represented by a stream of signed digits −1, 0, 1, a digit d representing the mapping x → (x + d)/2. This has been generalized to linear fractional representations studied in Edalat and Sunderhauf [11] as well as Edalat and Heckmann [9] where digits represent maps of the form x →/(cx+d)
A variant of the signed digit representation is the infinite Gray code introduced by Tsuiki [14] which represents real numbers in [−1, 1] by a binary stream with possibly one undefined entry
Summary
Digital representations of real numbers have been widely studied in the literature. Probably best known is the signed digit representation as considered in Ciaffaglione and Di Gianantonio [7], Escardoand Marcel-Romero [12] Berger and Hou [4] and many others, where a real number in [−1, 1] is represented by a stream of signed digits −1, 0, 1, a digit d representing the mapping x → (x + d)/2. We introduce a coinductive predicate on the powerset of X whose realizers are trees representing nonempty compact subsets of X generalizing the coinductive approach to the signed digit representation studied by Berger [2]. We sketch how this approach can be used to extract programs computing with compact sets from constructive proofs and comment on the relation to iterated function systems as studied by Edalat [8].
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