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Program extraction in exact real arithmetic†

Published online by Cambridge University Press:  10 November 2014

KENJI MIYAMOTO  and
HELMUT SCHWICHTENBERG
KENJI MIYAMOTO
Affiliation:
Mathematisches Institut, LMU, Theresienstr. 39, D-80333 München, Germany Email: miyamoto@math.lmu.de, schwicht@math.lmu.de
HELMUT SCHWICHTENBERG
Affiliation:
Mathematisches Institut, LMU, Theresienstr. 39, D-80333 München, Germany Email: miyamoto@math.lmu.de, schwicht@math.lmu.de
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Abstract

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The importance of an abstract approach to a computation theory over general data types has been stressed by Tucker in many of his papers. Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machine extract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account.

Footnotes

†

Kenji Miyamoto is supported by the Marie Curie Initial Training Network in Mathematical Logic – MALOA – from Mathematical Logic to Applications, PITN-GA-2009-238381

References

Agda. (2013) Available at http://wiki.portal.chalmers.se/agda/ Google Scholar
Berger, U. (1993) Program extraction from normalization proofs. In: Bezem, M. and Groote, J. (eds.) Typed Lambda Calculi and Applications. Springer Verlag Lecture Notes in Computer Science 664 91–106.CrossRefGoogle Scholar
Berger, U. (2009) From coinductive proofs to exact real arithmetic. In: Grädel, E. and Kahle, R. (eds.) Computer Science Logic. Springer Verlag Lecture Notes in Computer Science 5771 132–146.Google Scholar
Berger, U., Eberl, M. and Schwichtenberg, H. (2003) Term rewriting for normalization by evaluation. Information and Computation 183 19–42.CrossRefGoogle Scholar
Berger, U. and Seisenberger, M. (2010) Proofs, programs, processes. In: Ferreira, F. et al. (eds.) Proceedings CiE 2010. Springer Verlag Lecture Notes Computer Science 6158 39–48.Google Scholar
Berger, U. and Seisenberger, M. (2012) Proofs, programs, processes. Theory of Computing 51 313–329.Google Scholar
Berghofer, S. (2003) Proofs, Programs and Executable Specifications in Higher Order Logic, Ph.D. thesis, Institut für Informatik, TU München.Google Scholar
Chuang, C. M. (2011) Extraction of Programs for Exact Real Number Computation Using Agda, Ph.D. thesis, Swansea University, Wales, UK.Google Scholar
Ciaffaglione, A. and Gianantonio, P. D. (2006) A certified, corecursive implementation of exact real numbers. Theoretical Computer Science 351 39–51.Google Scholar
Coq Development Team (2009) The Coq Proof Assistant Reference Manual – Version 8.2. Inria.Google Scholar
Kohlenbach, U. (2008) Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Verlag, Berlin, Heidelberg.Google Scholar
Kreisel, G. (1959) Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam, 101–128.Google Scholar
Letouzey, P. (2003) A new extraction for coq. In: Geuvers, H. and Wiedijk, F. (eds.) Types for Proofs and Programs, Second International Workshop, TYPES 2002. Springer Verlag Lecture Notes in Computer Science 2646 200–219.Google Scholar
O'Connor, R. (2009) Incompleteness & Completeness. Formalizing Logic and Analysis in Type Theory, Ph.D. thesis, Nijmegen University.Google Scholar
Plume, D. (1998) A Calculator for Exact Real Number Computation, Ph.D. thesis, University of Edinburgh.Google Scholar
Schwichtenberg, H. (2006) Minlog. In: Wiedijk, F. (ed.) The Seventeen Provers of the World. Springer Verlag Lecture Notes in Artificial Intelligence 3600 151–157.Google Scholar
Schwichtenberg, H. (2008) Realizability interpretation of proofs in constructive analysis. Theory of Computing Systems 43 (3) 583–602.Google Scholar
Schwichtenberg, H. and Wainer, S. S. (2012) Proofs and Computations. Perspectives in Logic. Cambridge University Press.Google Scholar
Tucker, J. V. and Zucker, J. I. (1992) Theory of computation over stream algebras, and its applications. In: Havel, I. M. and Koubek, V. (eds.) Mathematical Foundations of Computer Science. Springer Verlag Lecture Notes in Computer Science 629 62–80.Google Scholar
Wiedmer, E. (1977) Exaktes Rechnen mit reellen Zahlen und anderen unendlichen Objekten, Ph.D. thesis, ETH Zürich.Google Scholar
Wiedmer, E. (1980) Computing with infinite objects. Theoretical Computer Science 10 133–155.Google Scholar