Limits of real numbers in the binary signed digit representation

TL;DR

This paper formalizes and verifies algorithms for exact real number computation using binary signed digit streams, demonstrating the closure of reals under limits and applying the results to compute square roots and multiplication.

Contribution

It provides a constructive proof of the closure of real numbers under limits using coinductive binary signed digit streams, with verified algorithms extracted in Haskell.

Findings

Verified algorithms for real number limits in binary signed digit representation

Proof of closure of reals under limits using two approaches

Algorithms for square root and multiplication in the signed digit representation

Abstract

We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams of binary signed digits. The main objective of this paper is the formalisation of a constructive proof that real numbers are closed with respect to limits. All the proofs of the main theorem and the first application are implemented in the Minlog proof system and the extracted terms are further translated into Haskell. We compare two approaches. The first approach is a direct proof. In the second approach we make use of the representation of reals by a Cauchy-sequence of rationals. Utilizing translations between the two represenation and using the completeness of the Cauchy-reals, the proof is very short. In both cases we use Minlog's program extraction mechanism to automatically extract a formally verified program that transforms a converging sequence of reals, i.e.~a sequence of streams of binary signed digits together with a modulus of convergence, into the binary signed digit representation of its limit. The correctness of the extracted terms follows directly from the soundness theorem of program extraction. As a first application we use the extracted algorithms together with Heron's method to construct an algorithm that computes square roots with respect to the binary signed digit representation. In a second application we use the convergence theorem to show that the signed digit representation of real numbers is closed under multiplication.