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A monadic, functional implementation of real numbers

Published online by Cambridge University Press:  01 February 2007

RUSSELL ÓCONNOR
RUSSELL ÓCONNOR*
Affiliation:
Institute for Computing and Information Science, Faculty of Science, Radboud UniversityNijmegen Email: r.oconnor@cs.ru.nl
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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.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 1 , February 2007 , pp. 129 - 159
Copyright
Copyright © Cambridge University Press 2007

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References

Bishop, E. and Bridges, D. (1985) Constructive Analysis. Grundlehren der mathematischen Wissenschaften 279, Springer-VerlagGoogle Scholar
Boehm, H.-J., Cartwright, R., Riggle, M. and O'Donnell, M. J. (1986) Exact real arithmetic: a case study in higher order programming. In: LFP '86: Proceedings of the 1986 ACM conference on LISP and functional programming, ACM Press 162–173.CrossRefGoogle Scholar
Burago, D., Burago, Y. and Ivanov, S. (2001) A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society.Google Scholar
Cruz-Filipe, L. (2003) A constructive formalization of the fundamental theorem of calculus. In: Geuvers, H. and Wiedijk, F. (eds.) Types for Proofs and Programs. Springer-Verlag Lecture Notes in Computer Science 2646 108–126.CrossRefGoogle Scholar
Cruz-Filipe, L. and Spitters, B. (2003) Program extraction from large proof developments. In: Basin, D. and Wolff, B. (eds.) Theorem Proving in Higher Order Logics, 16th International Conference, TPHOLs 2003. Springer-Verlag Lecture Notes in Computer Science 2758205–220.Google Scholar
Hales, T. C. (2002) A computer verification of the Kepler conjecture. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, Volume III, Higher Ed. Press 795–804.Google Scholar
Kurzweg, U. H. (2001) Pi. (Available at http://aemes.mae.ufl.edu/~uhk/PI.html.)Google Scholar
Lester, D. (2003) Using PVS to validate the inverse trigonometric functions of an exact arithmetic. In: Alt, R., Frommer, A., Kearfott, R. B. and Luther, W. (eds.) Numerical Software with Result Verification. Springer-Verlag Lecture Notes in Computer Science 2991259–273.Google Scholar
Lester, D. and Gowland, P. (2002) Using PVS to validate the algorithms of an exact arithmetic. Theoretical Computer Science 291 (2)203–218.CrossRefGoogle Scholar
Moggi, E. (1991) Notions of computation and monads. Information and Computation 93 (1)55–92.CrossRefGoogle Scholar
Muñoz, C. and Lester, D. (2005) Real number calculations and theorem proving. In: Hurd, J. and Melham, T. (eds.) Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005. Springer-Verlag Lecture Notes in Computer Science 3603195–210.Google Scholar
Niqui, M. and Wiedijk, F. (2005) The ‘many digits’ friendly competition 2005. (Details at http://www.cs.ru.nl/~milad/manydigits.)Google Scholar
O'Connor, R. (2005) Few digits 0.4.0. (Available at http://r6.ca/FewDigits/.)Google Scholar
Odlyzko, A. M. and te Riele, H. J. J. (1985) Disproof of the Mertens conjecture. J. Reine Angew. Math. 357 138–160.Google Scholar
Peyton Jones, S. (ed.) (2003) Haskell 98 Language and Libraries: The Revised Report, Cambridge University Press.Google Scholar
The Coq development team (2004) The Coq proof assistant reference manual, Version 8.0, LogiCal Project.Google Scholar
Wadler, P. (1992) Monads for functional programming. In: Proceedings of the Marktoberdorf Summer School on Program Design Calculi.Google Scholar