<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of hal-04664615</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-25T05:00:42+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">The Complexity of Computing in Continuous Time: Space Complexity Is Precision</title>
            <author role="aut">
              <persName>
                <forename type="first">Manon</forename>
                <surname>Blanc</surname>
              </persName>
              <email type="md5">334a7b3b608d6dd4a7e3233ed6f34273</email>
              <email type="domain">lix.polytechnique.fr</email>
              <idno type="idhal" notation="string">manon-blanc</idno>
              <idno type="idhal" notation="numeric">1261491</idno>
              <idno type="halauthorid" notation="string">2634236-1261491</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-6961-089X</idno>
              <idno type="IDREF">https://www.idref.fr/285447289</idno>
              <affiliation ref="#struct-2071"/>
              <affiliation ref="#struct-1061259"/>
            </author>
            <author role="aut">
              <persName>
                <forename type="first">Olivier</forename>
                <surname>Bournez</surname>
              </persName>
              <email type="md5">66d34bc2caca9db97f864c77209305ce</email>
              <email type="domain">lix.polytechnique.fr</email>
              <idno type="idhal" notation="string">olivierbournez</idno>
              <idno type="idhal" notation="numeric">754539</idno>
              <idno type="halauthorid" notation="string">9604-754539</idno>
              <idno type="ORCID">https://orcid.org/0000-0002-9218-1130</idno>
              <idno type="IDREF">https://www.idref.fr/136239927</idno>
              <affiliation ref="#struct-2071"/>
              <affiliation ref="#struct-300340"/>
              <affiliation ref="#struct-563936"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Manon</forename>
                <surname>Blanc</surname>
              </persName>
              <email type="md5">4b800cfba05fa10b158c8ecd54d558c8</email>
              <email type="domain">crans.org</email>
            </editor>
          </titleStmt>
          <editionStmt>
            <edition n="v1" type="current">
              <date type="whenSubmitted">2024-07-30 12:49:53</date>
              <date type="whenModified">2025-04-10 11:48:09</date>
              <date type="whenReleased">2024-07-31 08:04:52</date>
              <date type="whenProduced">2024-07-08</date>
              <date type="whenEndEmbargoed">2024-07-30</date>
              <ref type="file" target="https://hal.science/hal-04664615v1/document">
                <date notBefore="2024-07-30"/>
              </ref>
              <ref type="file" subtype="greenPublisher" n="1" target="https://hal.science/hal-04664615v1/file/LIPIcs.ICALP.2024.129.pdf" id="file-4664617-4050637">
                <date notBefore="2024-07-30"/>
              </ref>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="1167789">
                <persName>
                  <forename>Manon</forename>
                  <surname>Blanc</surname>
                </persName>
                <email type="md5">4b800cfba05fa10b158c8ecd54d558c8</email>
                <email type="domain">crans.org</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">hal-04664615</idno>
            <idno type="halUri">https://hal.science/hal-04664615</idno>
            <idno type="halBibtex">blanc:hal-04664615</idno>
            <idno type="halRefHtml">&lt;i&gt;International Colloquium on Automata, Languages, and Programming (ICALP)&lt;/i&gt;, Jul 2024, Tallinn, Estonia. &lt;a target="_blank" href="https://dx.doi.org/10.4230/LIPIcs.ICALP.2024.129"&gt;&amp;#x27E8;10.4230/LIPIcs.ICALP.2024.129&amp;#x27E9;&lt;/a&gt;</idno>
            <idno type="halRef">International Colloquium on Automata, Languages, and Programming (ICALP), Jul 2024, Tallinn, Estonia. &amp;#x27E8;10.4230/LIPIcs.ICALP.2024.129&amp;#x27E9;</idno>
            <availability status="restricted">
              <licence target="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0 - Attribution<ref corresp="#file-4664617-4050637"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt>
            <idno type="stamp" n="X">École polytechnique</idno>
            <idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
            <idno type="stamp" n="LIX" corresp="X">Laboratoire d'informatique de l'Ecole polytechnique</idno>
            <idno type="stamp" n="X-DEP-INFO">Département d'informatique de l’École polytechnique</idno>
            <idno type="stamp" n="CENTRALESUPELEC">Ecole CentraleSupélec</idno>
            <idno type="stamp" n="UNIV-PARIS-SACLAY">Université Paris-Saclay</idno>
            <idno type="stamp" n="IP_PARIS">Institut Polytechnique de Paris</idno>
            <idno type="stamp" n="UNIVERSITE-PARIS-SACLAY" corresp="UNIV-PARIS-SACLAY">Université Paris-Saclay</idno>
            <idno type="stamp" n="LISN">Laboratoire Interdisciplinaire des Sciences du Numérique</idno>
            <idno type="stamp" n="GS-COMPUTER-SCIENCE">Graduate School Computer Science</idno>
            <idno type="stamp" n="LISN-GALAC" corresp="LISN">Graphes, Algorithmes et Combinatoire </idno>
            <idno type="stamp" n="PSACLAY-TEST">Université Paris-Saclay - Collection pour tests</idno>
          </seriesStmt>
          <notesStmt>
            <note type="audience" n="2">International</note>
            <note type="invited" n="0">No</note>
            <note type="popular" n="0">No</note>
            <note type="peer" n="1">Yes</note>
            <note type="proceedings" n="1">Yes</note>
          </notesStmt>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">The Complexity of Computing in Continuous Time: Space Complexity Is Precision</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Manon</forename>
                    <surname>Blanc</surname>
                  </persName>
                  <email type="md5">334a7b3b608d6dd4a7e3233ed6f34273</email>
                  <email type="domain">lix.polytechnique.fr</email>
                  <idno type="idhal" notation="string">manon-blanc</idno>
                  <idno type="idhal" notation="numeric">1261491</idno>
                  <idno type="halauthorid" notation="string">2634236-1261491</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-6961-089X</idno>
                  <idno type="IDREF">https://www.idref.fr/285447289</idno>
                  <affiliation ref="#struct-2071"/>
                  <affiliation ref="#struct-1061259"/>
                </author>
                <author role="aut">
                  <persName>
                    <forename type="first">Olivier</forename>
                    <surname>Bournez</surname>
                  </persName>
                  <email type="md5">66d34bc2caca9db97f864c77209305ce</email>
                  <email type="domain">lix.polytechnique.fr</email>
                  <idno type="idhal" notation="string">olivierbournez</idno>
                  <idno type="idhal" notation="numeric">754539</idno>
                  <idno type="halauthorid" notation="string">9604-754539</idno>
                  <idno type="ORCID">https://orcid.org/0000-0002-9218-1130</idno>
                  <idno type="IDREF">https://www.idref.fr/136239927</idno>
                  <affiliation ref="#struct-2071"/>
                  <affiliation ref="#struct-300340"/>
                  <affiliation ref="#struct-563936"/>
                </author>
              </analytic>
              <monogr>
                <title level="m">Leibniz International Proceedings in Informatics (LIPIcs)</title>
                <meeting>
                  <title>International Colloquium on Automata, Languages, and Programming (ICALP)</title>
                  <date type="start">2024-07-08</date>
                  <date type="end">2024-07-12</date>
                  <settlement>Tallinn</settlement>
                  <country key="EE">Estonia</country>
                </meeting>
                <imprint>
                  <publisher>Schloss Dagstuhl – Leibniz-Zentrum für Informatik</publisher>
                  <biblScope unit="volume">51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)</biblScope>
                  <date type="datePub">2024</date>
                </imprint>
              </monogr>
              <idno type="doi">10.4230/LIPIcs.ICALP.2024.129</idno>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <keywords scheme="author">
                <term xml:lang="en">Models of computation</term>
                <term xml:lang="en">Ordinary differential equations</term>
                <term xml:lang="en">Real computations</term>
                <term xml:lang="en">Analog computations</term>
                <term xml:lang="en">Complexity theory</term>
                <term xml:lang="en">Implicit complexity</term>
                <term xml:lang="en">Recursion scheme</term>
                <term xml:lang="en">Theory of computation → Models of computation</term>
                <term xml:lang="en">Theory of computation → Computability</term>
                <term xml:lang="en">Theory of computation → Complexity classes</term>
                <term xml:lang="en">Mathematics of computing → Ordinary differential equations</term>
              </keywords>
              <classCode scheme="halDomain" n="info">Computer Science [cs]</classCode>
              <classCode scheme="halTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halOldTypology" n="COMM">Conference papers</classCode>
              <classCode scheme="halTreeTypology" n="COMM">Conference papers</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>Models of computations over the integers are equivalent from a computability and complexity theory point of view by the (effective) Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but simpler for continuous-time models, as there is a unifying mathematical model, provided by ordinary differential equations (ODEs). Each model corresponds to a particular class of ODEs. For example, the General Purpose Analog Computer model of Claude Shannon, introduced as a mathematical model of analogue machines (Differential Analyzers), is known to correspond to polynomial ODEs. However, the question of a robust complexity theory for such models and its relations to classical (discrete) computation theory is an old problem. There was some recent significant progress: it has been proved that (classical) time complexity corresponds to the length of the involved curves, i.e. to the length of the solutions of the corresponding polynomial ODEs. The question of whether there is a simple and robust way to measure space complexity remains. We argue that space complexity corresponds to precision and conversely. Concretely, we propose and prove an algebraic characterisation of FPSPACE, using continuous ODEs. Recent papers proposed algebraic characterisations of polynomial-time and polynomial-space complexity classes over the reals, but with a discrete-time: those algebras rely on discrete ODE schemes. Here, we use classical (continuous) ODEs, with the classic definition of derivation and hence with the more natural context of continuous-time associated with ODEs. We characterise both the case of polynomial space functions over the integers and the reals. This is done by proving two inclusions. The first is obtained using some original polynomial space method for solving ODEs. For the other, we prove that Turing machines, with a proper representation of real numbers, can be simulated by continuous ODEs and not just discrete ODEs. A major consequence is that the associated space complexity is provably related to the numerical stability of involved schemas and the associated required precision. We obtain that a problem can be solved in polynomial space if and only if it can be simulated by some numerically stable ODE, using a polynomial precision.</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="laboratory" xml:id="struct-2071" status="VALID">
          <idno type="IdRef">196509955</idno>
          <idno type="ISNI">0000000403673665</idno>
          <idno type="RNSR">200519331V</idno>
          <idno type="ROR">https://ror.org/04afed728</idno>
          <idno type="Wikidata">Q16009025</idno>
          <orgName>Laboratoire d'informatique de l'École polytechnique [Palaiseau]</orgName>
          <orgName type="acronym">LIX</orgName>
          <date type="start">2005-01-01</date>
          <desc>
            <address>
              <addrLine>1 Rue Honoré d’Estienne d’Orves, Bâtiment Alain Turing, 91120 Palaiseau</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.lix.polytechnique.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-300340" type="direct"/>
            <relation active="#struct-563936" type="indirect"/>
            <relation name="UMR7161" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="laboratory" xml:id="struct-1061259" status="VALID">
          <idno type="IdRef">253160316</idno>
          <idno type="RNSR">202123711L</idno>
          <idno type="ROR">https://ror.org/00rd81916</idno>
          <idno type="Wikidata">Q109620676</idno>
          <orgName>Laboratoire Interdisciplinaire des Sciences du Numérique</orgName>
          <orgName type="acronym">LISN</orgName>
          <date type="start">2021-01-01</date>
          <desc>
            <address>
              <addrLine>Campus Universitaire bât 507Rue du Belvédère91405 Orsay cedexet Campus Universitaire bât 6401 rue Raimond Castaing91190 Gif-sur-Yvette</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.lisn.upsaclay.fr/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-300009" type="direct"/>
            <relation active="#struct-411575" type="direct"/>
            <relation active="#struct-419361" type="direct"/>
            <relation name="UMR9015" active="#struct-441569" type="direct"/>
          </listRelation>
        </org>
        <org type="institution" xml:id="struct-300340" status="VALID">
          <idno type="IdRef">027309320</idno>
          <idno type="ISNI">0000000121581279</idno>
          <idno type="ROR">https://ror.org/05hy3tk52</idno>
          <idno type="Wikidata">Q273626</idno>
          <orgName>École polytechnique</orgName>
          <orgName type="acronym">X</orgName>
          <date type="start">1794-03-11</date>
          <desc>
            <address>
              <addrLine>Route de Saclay, 91128 Palaiseau Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.polytechnique.edu/</ref>
          </desc>
          <listRelation>
            <relation active="#struct-563936" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-563936" status="VALID">
          <idno type="IdRef">238327159</idno>
          <idno type="ISNI">0000000502717600</idno>
          <idno type="ROR">https://ror.org/042tfbd02</idno>
          <idno type="Wikidata">Q48759778</idno>
          <orgName>Institut Polytechnique de Paris</orgName>
          <orgName type="acronym">IP Paris</orgName>
          <date type="start">2019-06-02</date>
          <desc>
            <address>
              <addrLine>Route de Saclay, 91120 Palaiseau Cedex, France</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.ip-paris.fr</ref>
          </desc>
        </org>
        <org type="regroupinstitution" xml:id="struct-441569" status="VALID">
          <idno type="IdRef">02636817X</idno>
          <idno type="ISNI">0000000122597504</idno>
          <idno type="ROR">https://ror.org/02feahw73</idno>
          <orgName>Centre National de la Recherche Scientifique</orgName>
          <orgName type="acronym">CNRS</orgName>
          <date type="start">1939-10-19</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.cnrs.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-300009" status="VALID">
          <idno type="ROR">https://ror.org/02kvxyf05</idno>
          <orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
          <orgName type="acronym">Inria</orgName>
          <desc>
            <address>
              <addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.inria.fr/en/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-411575" status="VALID">
          <idno type="IdRef">184443237</idno>
          <idno type="ROR">https://ror.org/019tcpt25</idno>
          <orgName>CentraleSupélec</orgName>
          <desc>
            <address>
              <addrLine>3, rue Joliot Curie,Plateau de Moulon,91192 GIF-SUR-YVETTE Cedex</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www.centralesupelec.fr</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-419361" status="VALID">
          <idno type="IdRef">241345251</idno>
          <idno type="ROR">https://ror.org/03xjwb503</idno>
          <orgName>Université Paris-Saclay</orgName>
          <desc>
            <address>
              <addrLine>Bâtiment Bréguet, 3 Rue Joliot Curie 2e ét, 91190 Gif-sur-Yvette</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.universite-paris-saclay.fr/fr</ref>
          </desc>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>