Computer Science – Numerical Analysis
Scientific paper
2006-05-14
Russell O'Connor: A monadic, functional implementation of real numbers. Mathematical Structures in Computer Science 17(1): 129
Computer Science
Numerical Analysis
This paper is to appear in an upcoming issue of Mathematical Structures in Computer Science published by Cambridge University
Scientific paper
10.1017/S0960129506005871
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 of regular sequences is generalized to, what I call, regular functions which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (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 easily verify its correctness.
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