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Formally Computing with the Non-computable

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Connecting with Computability (CiE 2021)

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Abstract

Church–Turing computability, which is the standard notion of computation, is based on functions for which there is an effective method for constructing their values. However, intuitionistic mathematics, as conceived by Brouwer, extends the notion of effective algorithmic constructions by also admitting constructions corresponding to human experiences of mathematical truths, which are based on temporal intuitions. In particular, the key notion of infinitely proceeding sequences of freely chosen objects, known as free choice sequences, regards functions as being constructed over time. This paper describes how free choice sequences can be embedded in an implemented formal framework, namely the constructive type theory of the Nuprl proof assistant. Some broader implications of supporting such an extended notion of computability in a formal system are then discussed, focusing on formal verification and constructive mathematics.

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Notes

  1. 1.

    Variants of bar induction were shown to be compatible with constructive type theory, and used to enhance the logical functionality implemented by proof assistants [29, 32].

  2. 2.

    For simplicity, throughout this paper we focus on choice sequences of natural numbers.

  3. 3.

    For a survey of the status of Church Thesis in type-theory-based proof assistants see [20].

  4. 4.

    The extended framework described was formalized in Coq’s formalization of Nuprl’s constructive type theory [3, 27].

  5. 5.

    This simplified description omits many components of the system which are not relevant to the current paper.

  6. 6.

    See, e.g., [35, 37, 38] for discussions on the various types of restrictions.

  7. 7.

    Notable exceptions include, e.g., [36, 42].

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Acknowledgments

The author thanks Vincent Rahli, Robert Constable and Mark Bickford as the framework described in the paper is based on a joint ongoing work with them.

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  1. Department of Computer Science, Ben-Gurion University, Be’er Sheva, Israel

    Liron Cohen

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  1. University of Lille, Lille, France

    Liesbeth De Mol

  2. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

  3. University of Göttingen, Göttingen, Germany

    Florin Manea

  4. Ghent University, Ghent, Belgium

    David Fernández-Duque

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Cohen, L. (2021). Formally Computing with the Non-computable. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_12

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  • DOI: https://doi.org/10.1007/978-3-030-80049-9_12

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