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2021 | OriginalPaper | Chapter

Formally Computing with the Non-computable

Author : Liron Cohen

Published in: Connecting with Computability

Publisher: Springer International Publishing

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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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previous chapter A Tale of Optimizing the Space Taken by de Bruijn Graphs
next chapter Mapping Monotonic Restrictions in Inductive Inference
Footnotes
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
For simplicity, throughout this paper we focus on choice sequences of natural numbers.
 
3
For a survey of the status of Church Thesis in type-theory-based proof assistants see [20].
 
4
The extended framework described was formalized in Coq’s formalization of Nuprl’s constructive type theory [3, 27].
 
5
This simplified description omits many components of the system which are not relevant to the current paper.
 
6
See, e.g., [35, 37, 38] for discussions on the various types of restrictions.
 
7
Notable exceptions include, e.g., [36, 42].
 
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Metadata
Title
Formally Computing with the Non-computable
Author
Liron Cohen
Copyright Year
2021
Book
Connecting with Computability

Print ISBN: 978-3-030-80048-2

Electronic ISBN: 978-3-030-80049-9

Copyright Year: 2021

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

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