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But… Why not to have a syntax built on the same principles as those of semantic composition?

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syntactic categories a, an :np/n very :(n/n)/(n/n) young :n/n student, sonata :n plays :(np\s)/np

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A very young student plays a sonata np/n (n/n)/(n/n) n/n n (np\s)/np np/n n np np\s n/n n np s

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Reduction rules Right cancellation Left cancellation

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Definition of syntactic types Primitive types: ex: np, n, s… (a finite set) Complex types: if A and B are types, - A/B is a type - B\A is a type

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learning of categories Start : marie ::= np, marie dort ::= s Mariedort s nps

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Start : marie ::= np, marie dort ::= s Mariedort s npnp\ss

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dortprofondémentnp\snp\s

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np\s(np\s)\(np\s)np\s une femme dort profondément s np\s

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dortprofondémentnp\s np\s(np\s)\(np\s)np\s une femme dort profondément s np np\s

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dortprofondémentnp\s np\s(np\s)\(np\s)np\s une femme dort profondément s s/(np\s) np\s

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functional interprétation B/A or A\B : functions from A to B B/A A B C/B B/A C/A

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other rules «type raising»: associativity composition

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Natural Deduction / - elimination: /- introduction: A/BB A [B]i A i A/B

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[B] : hypothesis labelled n°i [B]i A i A/B : the hypothesis n°i is discharged

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Example: « type-raising » A [A\B]1 B B/(A\B) 1

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Le livre que Pierre lit sn/nn(n\n)/(s/sn)sn(sn\s)/sn[sn]1 sn\s s n\n n sn 1 s/sn

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but… natural deductions are precisely -terms !

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f : A/B :B f( ):A [x:B]i u:A i x.u:A/B

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Pierre lit Tintin p:sn :(sn\s)/snsn x. y.lit(y,x) t:t: x. y.lit(y,x)) (t): sn\s y.lit(y,t) y.lit(y,t))(p): s lit(p,t)

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Pierre lit un livre (Peter reads a book)

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un livre (a book)

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Pierre lit (Peter reads) p:sn x. y.lit(y, x): (sn\s)/sn [u: sn]1 y.lit(y, u): sn\s [v:sn]2 lit(v, u): s 1 u.lit(v,u):s/sn 2 v. u.lit(v,u):sn\(s/sn) u.lit(p,u):s/sn

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Pierre lit un livre u.lit(p, u): s/sn

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Curry-Howard deduction / or \ - elimination / or \ - introduction hypothesis discharged hypothesis normalisation -term application abstraction variable bound variable -reduction

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Normalisation and -reduction A natural deduction is said to be normal whenever it does not contain an introduction rule followed by an elimination rule: [A] B/A A B B

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Normalisation and -reduction A natural deduction is said to be normal whenever it does not contain an introduction rule followed by an elimination rule: [A] B/A B B A

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Normalisation and -reduction A natural deduction is said to be normal whenever it does not contain an introduction rule followed by an elimination rule: [A] B/A B B A B [A]

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Normalisation and -reduction A natural deduction is said to be normal whenever it does not contain an introduction rule followed by an elimination rule: B/A B [A] B A A B

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Normalisation and -reduction B/A B [A] B A A B ( x A. B A) B [ A /x A ]

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sequent calculus (intuitionist) sequent antecedent consequent

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To prove : A/B amounts to prove : B and then : A

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Lambek calculus (with product) (sequents)

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A fundamental restriction: non empty antecedents a simple exercise a very simple exercise *a very exercise npnn nnnn n,nnnnnn/),//()/(,/,/,/ nnn... nnnn //,/ nn,,, npnn nnnn nnnn),//()/(/ //... nn /

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What sequent calculus reveals to us… cf. classical logic (some rules) (note the symmetries)

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but also: (on the two sides) + axiom and cut-rule Permutation Contraction Weakening

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Lambek calculus = intuitionistic logic WITHOUT A, C, P Intuitionistic multiplicative linear logic (+ restriction on non empty antecedents)

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subformula property AABB B/A A/B A\B B\A AB A B

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le livre que Pierre lit (the book that Peter reads) lelivrequePierrelit

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But…cut-rule AA

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L {Cut} = L Fortunately : Cut-elimination theorem

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Labelled Lambek calculus f f( ) xu x.u

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Pierre lit Tintin (Pierre reads Tintin)

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Pierre lit Tintin

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General properties of Lambek grammars Weak equivalence with CFGs : –A result by M. Pentus (1993) No strong equivalence with CFGs : –A result by H. J. Tiede (1998) Polynomiality? No result yet… –probably NP complete

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Limitations They are numerous: –only peripheral extraction The girl who I met : OK The girl who I met yesterday (or on the beach) : not OK –coordination and polymorphic types The mathematician whom Gottlob admired and Kazimierz detested : OK *The mathematician whom Gottlob admired Jim and Kazimierz detested : also OK!

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–parasitic gaps The book John filed _ without reading _ (linearity properties) –empty signs The book John read (cf. non empty antecedents)

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Extensions Multimodal Categorial Grammar (Moortgat, Oehrle, Morrill and their students) –ref. Categorial Type Logics in Van Benthem and ter Meulen (HLL) see further…

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A « cousin » Minimalist Grammars: –Inspired by Chomskys minimalist program –Ed. Stabler they have also type-logical formulations: –W. Vermaat –Retoré – Lecomte see further… or another day…

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to sum up We get rid of « syntactic » rules… by means of a logic which accepts a natural deduction presentation (because intuitionist) –proofs are -terms and also a sequent calculus –convenient for the proof search a logic which is linear (resource sensitive)

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