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Uniform Envelopes

Eike Neumann

Logical Methods in Computer Science, Volume: 18, Issue: 3, Pages: 8:1 - 8:34

Swansea University Author: Eike Neumann

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DOI (Published version): 10.46298/lmcs-18(3:8)2022

Abstract

In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function f: X → Y between qcb₀-spaces one can assign a so-called universal envelope which, in a well-defined sense, encodes all...

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Published in: Logical Methods in Computer Science
ISSN: 1860-5974
Published: Centre pour la Communication Scientifique Directe (CCSD) 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa60705
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spelling 2022-08-26T15:48:34.1222995 v2 60705 2022-08-02 Uniform Envelopes 1bf535eaa8d6fcdfbd464a511c1c0c78 Eike Neumann Eike Neumann true false 2022-08-02 SCS In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function f: X → Y between qcb₀-spaces one can assign a so-called universal envelope which, in a well-defined sense, encodes all continuously obtainable information on the function. A universal envelope consists of two continuous functions F: X → L and ξL: Y → L with values in a Σ-split injective space L. Any continuous function with values in an injective space whose composition with the original function is again continuous factors through the universal envelope. However, it is not possible in general to uniformly compute this factorisation. In this paper we propose the notion of uniform envelopes. A uniform envelope is additionally endowed with a map uL: L → ²(Y) that is compatible with the multiplication of the double powerspace monad ² in a certain sense. This yields for every continuous map with values in an injective space a choice of uniformly computable extension. Under a suitable condition which we call uniform universality, this extension yields a uniformly computable solution for the above factorisation problem. Uniform envelopes can be endowed with a composition operation. We establish criteria that ensure that the composition of two uniformly universal envelopes is again uniformly universal. These criteria admit a partial converse and we provide evidence that they cannot be easily improved in general. Not every function admits a uniformly universal uniform envelope. We can however assign to every function a canonical envelope that is in some sense as close as possible to a uniform envelope. We obtain a composition theorem similar to the uniform case. Journal Article Logical Methods in Computer Science 18 3 8:1 8:34 Centre pour la Communication Scientifique Directe (CCSD) 1860-5974 computable analysis, discontinuous functions, envelopes, complete lattices 28 7 2022 2022-07-28 10.46298/lmcs-18(3:8)2022 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University Not Required This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 731143. 2022-08-26T15:48:34.1222995 2022-08-02T12:51:14.8111784 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Eike Neumann 1 60705__25028__ad382d696e9d474f8fa88ea95351fa4d.pdf 60705_VoR.pdf 2022-08-26T15:47:31.7421631 Output 542635 application/pdf Version of Record true This work is licensed under the Creative Commons Attribution License. true eng https://creativecommons.org/licenses/by/4.0/
title Uniform Envelopes
spellingShingle Uniform Envelopes
Eike Neumann
title_short Uniform Envelopes
title_full Uniform Envelopes
title_fullStr Uniform Envelopes
title_full_unstemmed Uniform Envelopes
title_sort Uniform Envelopes
author_id_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78
author_id_fullname_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78_***_Eike Neumann
author Eike Neumann
author2 Eike Neumann
format Journal article
container_title Logical Methods in Computer Science
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publishDate 2022
institution Swansea University
issn 1860-5974
doi_str_mv 10.46298/lmcs-18(3:8)2022
publisher Centre pour la Communication Scientifique Directe (CCSD)
college_str Faculty of Science and Engineering
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hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
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department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
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description In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function f: X → Y between qcb₀-spaces one can assign a so-called universal envelope which, in a well-defined sense, encodes all continuously obtainable information on the function. A universal envelope consists of two continuous functions F: X → L and ξL: Y → L with values in a Σ-split injective space L. Any continuous function with values in an injective space whose composition with the original function is again continuous factors through the universal envelope. However, it is not possible in general to uniformly compute this factorisation. In this paper we propose the notion of uniform envelopes. A uniform envelope is additionally endowed with a map uL: L → ²(Y) that is compatible with the multiplication of the double powerspace monad ² in a certain sense. This yields for every continuous map with values in an injective space a choice of uniformly computable extension. Under a suitable condition which we call uniform universality, this extension yields a uniformly computable solution for the above factorisation problem. Uniform envelopes can be endowed with a composition operation. We establish criteria that ensure that the composition of two uniformly universal envelopes is again uniformly universal. These criteria admit a partial converse and we provide evidence that they cannot be easily improved in general. Not every function admits a uniformly universal uniform envelope. We can however assign to every function a canonical envelope that is in some sense as close as possible to a uniform envelope. We obtain a composition theorem similar to the uniform case.
published_date 2022-07-28T04:19:02Z
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