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Verified Exact Real Computation with Nondeterministic Functions and Limits

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Fundamentals of Computation Theory (FCT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14292))

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Abstract

The problems of computing limit points nondeterministically from sequences of nondeterministic real numbers appear ubiquitously in exact real computation along with root-finding of real and complex functions. To provide a rigorous foundation of verified computations with nondeterministic limits, we introduce a simple imperative language for exact real computation with a nondeterministic limit operator as its primitive. The operator’s formal semantics is defined. To make nontrivial sequences of nondeterministic real numbers be defined within the language, we further extend the language with lambda expressions for constructing higher-order nondeterministic functions without side effects and countable nondeterministic choices. We devise proof rules for the new operations and prove their soundness. As an example, to demonstrate the strength of the proof rules, we verify the correctness of a program, a computational counterpart of a constructive Intermediate Value Theorem, computing nondeterministically a root of a continuous locally non-constant real function whose signs at each endpoint of the unit interval differ.

The author is a JSPS International Research Fellow supported by JSPS KAKENHI (Grant-in-Aid for JSPS Fellows) JP22F22071.

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Notes

  1. 1.

    It is also often called multivaluedness or non-extensionality; see [6].

  2. 2.

    Often, a different notation \(f :\subseteq X \rightrightarrows Y\) is used for a partial nondeterministic function and \(f : X \rightrightarrows Y\) denotes a total nondeterministic function. However, we do not make a syntactic distinction between total and partial nondeterministic functions, and assume that all nondeterministic functions are possibly partial. Hence, in this paper, we use \(X \rightrightarrows Y\) for the set of partial nondeterministic functions. For example, \((X \rightrightarrows Y) \rightrightarrows Z\) denotes the set of partial nondeterministic functions to Z from partial nondeterministic functions from X to Y.

  3. 3.

    In exact real computation, it is also called Kleenean; see [6].

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Acknowledgements

The author would like to thank Holger Thies and the anonymous reviewers for constructive comments on the manuscript.

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  1. Kyoto University, Kyoto, Japan

    Sewon Park

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  1. Sewon Park
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Correspondence to Sewon Park .

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  1. University of Trier, Trier, Germany

    Henning Fernau

  2. University of Kiel, Kiel, Germany

    Klaus Jansen

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Park, S. (2023). Verified Exact Real Computation with Nondeterministic Functions and Limits. In: Fernau, H., Jansen, K. (eds) Fundamentals of Computation Theory. FCT 2023. Lecture Notes in Computer Science, vol 14292. Springer, Cham. https://doi.org/10.1007/978-3-031-43587-4_26

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  • DOI: https://doi.org/10.1007/978-3-031-43587-4_26

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