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Proofs, Programs, Processes

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Abstract

The objective of this paper is to provide a theoretical foundation for program extraction from inductive and coinductive proofs geared to practical applications. The novelties consist in the addition of inductive and coinductive definitions to a realizability interpretation for first-order proofs, a soundness proof for this system, and applications to the synthesis of non-trivial provably correct programs in the area of exact real number computation. We show that realizers, although per se untyped, can be assigned polymorphic recursive types and hence represent valid programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. the binary signed digit representation.

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References

  1. Benl, H., Berger, U., Schwichtenberg, H., Seisenberger, M., Zuber, W.: Proof theory at work: Program development in the Minlog system. In: Bibel, W., Schmitt, P.H. (eds.) Automated Deduction—A Basis for Applications. Applied Logic Series, vol. II, pp. 41–71. Kluwer, Dordrecht (1998)

    Google Scholar 

  2. Berger, U.: From coinductive proofs to exact real arithmetic: theory and applications. Log. Methods Comput. Sci. 7(1), (2011)

  3. Bertot, Y.: Affine functions and series with co-inductive real numbers. Math. Struct. Comput. Sci. 17, 37–63 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, U.: From coinductive proofs to exact real arithmetic. In: Grädel, E., Kahle, R. (eds.) Computer Science Logic. Lecture Notes in Comput. Sci., vol. 5771 pp. 132–146. Springer, Berlin (2009)

    Chapter  Google Scholar 

  5. Berger, U.: Realisability for induction and coinduction with applications to constructive analysis. J. Univers. Comput. Sci. 16(18), 2535–2555 (2010)

    MATH  Google Scholar 

  6. Buchholz, W., Feferman, F., Pohlers, W., Sieg, W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof–Theoretical Studies. Lecture Notes in Mathematics, vol. 897. Springer, Berlin (1981)

    MATH  Google Scholar 

  7. Berger, U., Hou, T.: Coinduction for exact real number computation. Theory Comput. Syst. 43, 394–409 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berger, U., Seisenberger, M.: Proofs, programs, processes. In: Ferreira, F., Löwe, B., Mayordomo, E., Gomes, L.M. (eds.) Programs, Proofs, Processes, 6th Conference on Computability in Europe, CiE 2010, Ponta Delgada, Azores, Portugal, June 30–July 4, 2010. Proceedings of Lecture Notes in Comput. Sci., vol. 6158, pp. 39–48. (2010)

    Google Scholar 

  9. Ciaffaglione, A., Di Gianantonio, P.: A certified, corecursive implementation of exact real numbers. Theor. Comput. Sci. 351, 39–51 (2006)

    Article  MATH  Google Scholar 

  10. The Coq Proof Assistant. http://coq.inria.fr/

  11. Edalat, A., Heckmann, R.: Computing with real numbers: I. The LFT approach to real number computation; II. A domain framework for computational geometry. In: Barthe, G., Dybjer, P., Pinto, L., Saraiva, J. (eds.) Applied Semantics—Lecture Notes from the International Summer School, Caminha, Portugal, pp. 193–267. Springer, Berlin (2002)

    Google Scholar 

  12. Ghani, N., Hancock, P., Pattinson, D.: Continuous functions on final coalgebras. Electron. Notes Theor. Comput. Sci. 164, 141–155 (2006)

    Article  MathSciNet  Google Scholar 

  13. Geuvers, H., Niqui, M., Spitters, B., Wiedijk, F.: Constructive analysis, types and exact real numbers. Math. Struct. Comput. Sci. 17(1), 3–36 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hernest, M.D., Oliva, P.: Hybrid functional interpretations. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008: Logic and Theory of Algorithms. Lecture Notes in Comput. Sci., vol. 5028, pp. 251–260. Springer, Berlin (2008)

    Chapter  Google Scholar 

  15. Krivine, J.-L.: Dependent choice, ‘quote’ and the clock. Theor. Comput. Sci. 308, 259–276 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. The Minlog System. http://www.mathematik.uni-muenchen.de/~minlog/

  17. Miranda-Perea, F.: Realizability for monotone clausular (co)inductive definitions. Electron. Notes Theor. Comput. Sci. 123, 179–193 (2005)

    Article  MathSciNet  Google Scholar 

  18. Marcial-Romero, J.R., Escardo, M.H.: Semantics of a sequential language for exact real-number computation. Theor. Comput. Sci. 379(1–2), 120–141 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Niqui, M.: Coinductive formal reasoning in exact real arithmetic. Log. Methods Comput. Sci. 4(3–6), 1–40 (2008)

    MathSciNet  Google Scholar 

  20. Plume, D.: A calculator for exact real number computation. Master’s thesis, University of Edinburgh, 1998

  21. Ratiu, D., Trifonov, T.: Exploring the computational content of the infinite pigeonhole principle. J. Logic Comput. doi:10.1093/logcom/exq011

  22. Schwichtenberg, H.: Realizability interpretation of proofs in constructive analysis. Theory Comput. Syst. 43(3–4), 583–602 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tatsuta, M.: Realizability of monotone coinductive definitions and its application to program synthesis. In: Parikh, R. (ed.) Mathematics of Program Construction. Lecture Notes in Mathematics, vol. 1422, pp. 338–364. Springer, Berlin (1998)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Swansea University, Swansea, SA2 8PP, Wales, UK

    Ulrich Berger & Monika Seisenberger

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  1. Ulrich Berger
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  2. Monika Seisenberger
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Correspondence to Monika Seisenberger.

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Berger, U., Seisenberger, M. Proofs, Programs, Processes. Theory Comput Syst 51, 313–329 (2012). https://doi.org/10.1007/s00224-011-9325-8

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  • DOI: https://doi.org/10.1007/s00224-011-9325-8

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