Normalization and sub-formula property for Lambek with product and PCMLL -- Partially Commutative Multiplicative Linear Logic
Identifieur interne : 003617 ( Hal/Curation ); précédent : 003616; suivant : 003618Normalization and sub-formula property for Lambek with product and PCMLL -- Partially Commutative Multiplicative Linear Logic
Auteurs : Maxime Amblard [France] ; Christian Retoré [France]Source :
Abstract
This paper establishes the normalisation of natural deduction or lambda calculus formulation of Intuitionistic Non Commutative Logic --- which involves both commutative and non commutative connectives. This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Regarding Intuitionistic Non Commutative Logic with full entropy does not have up to now a proof net syntax, and that for linguistic applications, sequent calculi which are only more or less equivalent to natural deduction, are not convenient because they lack the standard Curry-Howard isomorphism.
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This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also includes, as a special case, the normalisation of natural deduction the Lambek calculus with product, which is unsurprising but yet unproved. Regarding Intuitionistic Non Commutative Logic with full entropy does not have up to now a proof net syntax, and that for linguistic applications, sequent calculi which are only more or less equivalent to natural deduction, are not convenient because they lack the standard Curry-Howard isomorphism.</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">Normalization and sub-formula property for Lambek with product and PCMLL -- Partially Commutative Multiplicative Linear Logic</title><author role="aut"><persName><forename type="first">Maxime</forename><surname>Amblard</surname></persName><email>maxime.amblard@loria.fr</email><ptr type="url" target="http://www.loria.fr/~amblard"></ptr><idno type="idhal">maxime-amblard</idno><idno type="halauthor">949515</idno><affiliation ref="#struct-150772"></affiliation></author><author role="aut"><persName><forename type="first">Christian</forename><surname>Retoré</surname></persName><email>christian.retore@labri.fr</email><ptr type="url" target="http://www.labri.fr/perso/retore"></ptr><idno type="halauthor">814204</idno><affiliation ref="#struct-3102"></affiliation></author><editor role="depositor"><persName><forename>Maxime</forename><surname>Amblard</surname></persName><email>maxime.amblard@loria.fr</email></editor></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2014-02-03 15:18:53</date><date type="whenWritten">2007-09-17</date><date type="whenModified">2015-09-21 11:22:39</date><date type="whenReleased">2014-02-03 20:26:05</date><date type="whenProduced">2007-09-17</date><date type="whenEndEmbargoed">2014-02-03</date><ref type="file" target="https://hal.archives-ouvertes.fr/hal-00941206/document"><date notBefore="2014-02-03"></date></ref><ref type="file" subtype="author" n="1" target="https://hal.archives-ouvertes.fr/hal-00941206/file/ambretcietorev.pdf"><date notBefore="2014-02-03"></date></ref></edition><respStmt><resp>contributor</resp><name key="122205"><persName><forename>Maxime</forename><surname>Amblard</surname></persName><email>maxime.amblard@loria.fr</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">hal-00941206</idno><idno type="halUri">https://hal.archives-ouvertes.fr/hal-00941206</idno><idno type="halBibtex">amblard:hal-00941206</idno><idno type="halRefHtml">2007</idno><idno type="halRef">2007</idno></publicationStmt><seriesStmt><idno type="stamp" n="CNRS">CNRS - 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