Limits with Signed Digit Streams

TL;DR

This paper develops algorithms for signed digit representations of real numbers, including limits and square roots, using constructive proofs and program extraction techniques with proof assistants and functional programming languages.

Contribution

It introduces a novel algorithm for limits of signed digit streams and applies it to compute square roots, using proof extraction from constructive proofs.

Findings

Algorithm for limits of signed digit streams

Constructive proofs enable direct extraction of algorithms

Implementation in Haskell demonstrates practical applicability

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.