Florian Steinberg ; Laurent Thery ; Holger Thies - Computable analysis and notions of continuity in Coq

lmcs:5418 - Logical Methods in Computer Science, May 12, 2021, Volume 17, Issue 2 - https://doi.org/10.23638/LMCS-17(2:16)2021
Computable analysis and notions of continuity in CoqArticle

Authors: Florian Steinberg ; Laurent Thery ; Holger Thies

    We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the paper can be used as an introduction to the library as it describes many of its most important features in detail. While the ability to have full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that certain countably infinite products are isomorphic to spaces of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also formalize proofs of non-computational results that support the correctness of our definitions. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented-space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers.

    Volume: Volume 17, Issue 2
    Published on: May 12, 2021
    Accepted on: October 29, 2020
    Submitted on: May 1, 2019
    Keywords: Computer Science - Logic in Computer Science
      Source : OpenAIRE Graph
    • Computing with Infinite Data; Funder: European Commission; Code: 731143
    • Fast Reliable Approximation; Funder: French National Research Agency (ANR); Code: ANR-14-CE25-0018

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