Mathematical Structures in Computer Science

Paper

A monadic, functional implementation of real numbers

RUSSELL ÓCONNORa1

a1 Institute for Computing and Information Science, Faculty of Science, Radboud University Nijmegen Email: r.oconnor@cs.ru.nl

Abstract

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion (Bishop and Bridges 1985) of regular sequences is generalised to what I call regular functions, which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (which are a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to verify its correctness easily.

(Received December 01 2005)

(Revised May 05 2006)