Limits with Signed Digit Streams

Abstract: We work with the signed digit representation of abstract real numbers, which roughly is the binary representation enriched by the additional digit -1. The main objective of this paper is an algorithm which takes a sequence of signed digit representations of reals and returns the signed digit representation of their limit, if the sequence converges. As a first application we use this algorithm together with Heron's method to build up an algorithm which converts the signed digit representation of a non-negative real number into the signed digit representation of its square root. Instead of writing the algorithms first and proving their correctness afterwards, we work the other way round, in the tradition of program extraction from proofs. In fact we first give constructive proofs, and from these proofs we then compute the extracted terms, which is the desired algorithm. The correctness of the extracted term follows directly by the Soundness Theorem of program extraction. In order to get the extracted term from some proofs which are often quite long, we use the proof assistant Minlog. However, to apply the extracted terms, the programming language Haskell is useful. Therefore after each proof we show a notation of the extracted term, which can be easily rewritten as a definition in Haskell.