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Parametrised second-order complexity theory with applications to the study of interval computation

Eike Neumann, Florian Steinberg

Theoretical Computer Science, Volume: 806, Pages: 281 - 304

Swansea University Author: Eike Neumann

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DOI (Published version): 10.1016/j.tcs.2019.05.009

Abstract

We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the...

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Published in: Theoretical Computer Science
ISSN: 0304-3975
Published: Elsevier BV 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa60142
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spelling 2022-10-31T14:34:09.5997647 v2 60142 2022-06-07 Parametrised second-order complexity theory with applications to the study of interval computation 1bf535eaa8d6fcdfbd464a511c1c0c78 Eike Neumann Eike Neumann true false 2022-06-07 SCS We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side.As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable. Journal Article Theoretical Computer Science 806 281 304 Elsevier BV 0304-3975 Second-order complexity, Type two complexity, Interval computation, Computable analysis 2 2 2020 2020-02-02 10.1016/j.tcs.2019.05.009 Author accepted manuscript available from: https://publications.aston.ac.uk/id/eprint/39143/ COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2022-10-31T14:34:09.5997647 2022-06-07T13:33:21.7830749 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Eike Neumann 1 Florian Steinberg 2
title Parametrised second-order complexity theory with applications to the study of interval computation
spellingShingle Parametrised second-order complexity theory with applications to the study of interval computation
Eike Neumann
title_short Parametrised second-order complexity theory with applications to the study of interval computation
title_full Parametrised second-order complexity theory with applications to the study of interval computation
title_fullStr Parametrised second-order complexity theory with applications to the study of interval computation
title_full_unstemmed Parametrised second-order complexity theory with applications to the study of interval computation
title_sort Parametrised second-order complexity theory with applications to the study of interval computation
author_id_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78
author_id_fullname_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78_***_Eike Neumann
author Eike Neumann
author2 Eike Neumann
Florian Steinberg
format Journal article
container_title Theoretical Computer Science
container_volume 806
container_start_page 281
publishDate 2020
institution Swansea University
issn 0304-3975
doi_str_mv 10.1016/j.tcs.2019.05.009
publisher Elsevier BV
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
document_store_str 0
active_str 0
description We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side.As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable.
published_date 2020-02-02T04:18:00Z
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score 11.013371

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