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

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

3. 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

4. Reduction rules Right cancellation Left cancellation

5. 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

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

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

8. dortprofondémentnp\snp\s

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

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

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

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

13. other rules «type raising»: associativity composition

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

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

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

17. 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

18. but… natural deductions are precisely -terms !

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

20. 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)

21. Pierre lit un livre (Peter reads a book)

22. un livre (a book)

23. 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

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

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

26. 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

27. 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

28. 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]

29. 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

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

31. sequent calculus (intuitionist) sequent antecedent consequent

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

33. Lambek calculus (with product) (sequents)

34. 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 /

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

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

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

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

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

46. But…cut-rule AA

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

48. Labelled Lambek calculus f f( ) xu x.u

49. Pierre lit Tintin (Pierre reads Tintin)

50. Pierre lit Tintin

55. 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

56. 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!

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

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

59. 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…

60. 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)