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Certified Computation of Nondeterministic Limits

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13260)


The computational content of constructive metric completeness is the operator that computes limits of Cauchy sequences. It can be used to construct certified programs that compute interesting transcendental real numbers from sequences of approximations. The desired nondeterministic version of it would be to nondeterministically compute real numbers from nondeterministic approximations. However, it is not obvious how nondeterministic metric completeness should be formalized.

We extend previous work on the formalization of exact real computation by primitive properties of nondeterminism. We show that by these properties, various forms of nondeterministic metric completeness can be derived without extending the axiomatic structure of constructive real numbers. We further implement our theory in the Coq proof assistant and use Coq’s code extraction features to extract efficient exact real computation programs using several forms of nondeterministic computation.


  • Constructive real numbers
  • Formal proofs
  • Exact real number computation
  • Program extraction
  • Nondeterminism

Holger Thies is supported by JSPS KAKENHI Grant Number JP20K19744. Sewon Park is supported by JSPS KAKENHI Grant number JP18H03203. This project has received funding from the EU’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 731143. The authors thank Franz Brauße and Norbert Müller for helpful discussions.

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  1. 1.

    in the sense of computable analysis [30].

  2. 2.

    Nondeterministic functions are also known as multivalued functions.


  1. Balluchi, A., Casagrande, A., Collins, P., Ferrari, A., Villa, T., Sangiovanni-Vincentelli, A.: Ariadne: a framework for reachability analysis of hybrid automata. In: Proceedings 17th International Symposium on Mathematical Theory of Networks and Systems. Kyoto (2006)

    Google Scholar 

  2. Berger, U., Tsuiki, H.: Intuitionistic fixed point logic. Ann. Pure Appl. Log. 172(3), 102903 (2021).

  3. Bishop, E.A.: Foundations of Constructive Analysis (1967)

    Google Scholar 

  4. Boldo, S., Melquiond, G.: Computer Arithmetic and Formal Proofs - Verifying Floating-point Algorithms with the Coq System. ISTE Press (2017).

  5. Brattka, V.: The emperor’s new recursiveness: the epigraph of the exponential function in two models of computability. In: Ito, M., Imaoka, T. (eds.) Words, Languages & Combinatorics III, pp. 63–72. World Scientific Publishing, Singapore (2003), iCWLC 2000, Kyoto, Japan, 14–18 March 2000

    Google Scholar 

  6. Brattka, V., Hertling, P.: Feasible real random access machines. J. Complex. 14(4), 490–526 (1998).,

  7. Bridges, D.S.: Constructive mathematics: a foundation for computable analysis. Theor. Comput. Sci. 219(1), 95–109 (1999).,

  8. Collins, P., Geretti, L., Casagrande, A., Zapreev, I., Zivanovic, S.: Ariadne (2005–20).

  9. Cooke, R.L.: Classical Algebra: its Nature, Origins, and Uses. John Wiley & Sons (2008)

    Google Scholar 

  10. Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-CoRN, the constructive coq repository at Nijmegen. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 88–103. Springer, Heidelberg (2004).

    CrossRef  Google Scholar 

  11. Brausse, F., Norbert Müller, R.R.: Intensionality and multi-valued limits. In: Proceedings 15th International Conference on Computability and Complexity in Analysis (CCA), p. 11 (2018)

    Google Scholar 

  12. Hertling, P.: A real number structure that is effectively categorical. Math. Log. Q. 45, 147–182 (1999).

  13. Hofmann, M.: On the interpretation of type theory in locally cartesian closed categories. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 427–441. Springer, Heidelberg (1995).

    CrossRef  Google Scholar 

  14. Konečný, M., Park, S., Thies, H.: Axiomatic reals and certified efficient exact real computation. In: Silva, A., Wassermann, R., de Queiroz, R. (eds.) WoLLIC 2021. LNCS, vol. 13038, pp. 252–268. Springer, Cham (2021).

    CrossRef  MATH  Google Scholar 

  15. Konečný, M.: Verified exact real limit computation. In: Proceedings 15th International Conference on Computability and Complexity in Analysis (CCA), pp. 9–10 (2018)

    Google Scholar 

  16. Konečný, M.: aern2-real: A Haskell library for exact real number computation. (2021)

  17. Letouzey, P.: A new extraction for Coq. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 200–219. Springer, Heidelberg (2003).

    CrossRef  MATH  Google Scholar 

  18. Letouzey, P.: Extraction in Coq: an overview. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 359–369. Springer, Heidelberg (2008).

    CrossRef  Google Scholar 

  19. Luckhardt, H.: A fundamental effect in computations on real numbers. Theor. Comput. Sci. 5(3), 321 – 324 (1977).,

  20. Müller, N.T.: Implementing limits in an interactive realram. In: 3rd Conference on Real Numbers and Computers, 1998, Paris. vol. 13, p. 26 (1998)

    Google Scholar 

  21. Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blanck, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001).

    CrossRef  Google Scholar 

  22. Müller, N.T., Uhrhan, C.: Some steps into verification of exact real arithmetic. In: Goodloe, A.E., Person, S. (eds.) NFM 2012. LNCS, vol. 7226, pp. 168–173. Springer, Heidelberg (2012).

    CrossRef  Google Scholar 

  23. Neumann, E., Pauly, A.: A topological view on algebraic computation models. J. Complex. 44, 1–22 (2018)

    CrossRef  MathSciNet  Google Scholar 

  24. Park, S., et al.: Foundation of computer (algebra) analysis systems: Semantics, logic, programming, verification. arXiv e-prints pp. arXiv-1608 (2016)

    Google Scholar 

  25. Reus, B.: Realizability models for type theories. Electron. Notes Theor. Comput. Sci. 23(1), 128–158 (1999)

    CrossRef  Google Scholar 

  26. Seely, R.A.G.: Locally cartesian closed categories and type theory. Math. Proc. Camb. Philoso. Soc. 95(1), 33–48 (1984).

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. Specker, E.: Nicht konstruktiv beweisbare Sätze der analysis. J. Symb. Logic 14(3), 145–158 (1949)

    CrossRef  Google Scholar 

  28. Steinberg, F., Thery, L., Thies, H.: Computable analysis and notions of continuity in Coq. Logical Meth. Comput. Sci. 17(2) (2021).

  29. Univalent Foundations Program, T.: Homotopy Type Theory: Univalent Foundations of Mathematics., Institute for Advanced Study (2013)

  30. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000).

    CrossRef  MATH  Google Scholar 

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Konečný, M., Park, S., Thies, H. (2022). Certified Computation of Nondeterministic Limits. In: Deshmukh, J.V., Havelund, K., Perez, I. (eds) NASA Formal Methods. NFM 2022. Lecture Notes in Computer Science, vol 13260. Springer, Cham.

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