But…

Why not to have a syntax built on the same principles as those of

semantic composition?

syntactic categories

• a, an : np/n

• very : (n/n)/(n/n)

• young : n/n

• student, sonata : n

• plays : (np\s)/np

A very young student plays a sonata

np/n(n/n)/(n/n)

n/nn

(np\s)/npnp/n n

np

np\sn/n

n

np

s

Reduction rules

• Right cancellation

• Left cancellation

ABBA /

AABB \

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

learning of categories

• Start : ‘marie’ ::= np, ‘marie dort’ ::= s

• Marie dort s

np s

• Start : ‘marie’ ::= np, ‘marie dort’ ::= s

• Marie dort s

np np\s s

• dort profondément np\s

np\s np\s

• dort profondément np\s

np\s (np\s)\(np\s) np\s

• une femme dort profondément s

np\s

• dort profondément np\s

np\s (np\s)\(np\s) np\s

• une femme dort profondément s

np np\s

• dort profondément np\s

np\s (np\s)\(np\s) np\s

• une femme dort profondément s

s/(np\s) np\s

functional interprétation

B/A or A\B : functions from A to B

BxfalorsAxetBAf

,:

CAfgalorsCBgetBAf :,::

B/A A B

C/B B/A C/A

other rules

• «type raising»:

• associativity

• composition

)\/( BABA BABA \)/(

)/(\/)\( CBACBA

CACBBA /// ACABBC \\\

Natural Deduction

/ - elimination:

/- introduction:

A/B B

A

[B]i

Ai

A/B

[B] : hypothesis labelled n°i

[B]i

Ai

A/B : the hypothesis n°i is discharged

Example:« type-raising »

)\(/ BABA

A [A\B]1

B

B/(A\B)1

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

sn\s

s

n\n

nsn

1s/sn

but…

• natural deductions are precisely -terms !

f : A/B :B

f():A

[x:B]i

u:A ix.u:A/B

Pierre lit Tintin

p:sn :(sn\s)/sn snx.y.lit(y,x) t:

x.y.lit(y,x))(t): sn\sy.lit(y,t)

y.lit(y,t))(p): slit(p,t)

Pierre lit un livre(Peter reads a book)

ns)/\sn)/((s:..

xvxuxvuun

n:. xlivrexlivre

ns)/\sn)/((s:..

xvxuxvu n:. xlivrex

un livre(a book)

s\sn)/(s:..

xvxxlivrexxv

s\sn)/(s:.

xvxlivrexv

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): s1

u.lit(v,u):s/sn2

v.u.lit(v,u):sn\(s/sn)

u.lit(p,u):s/sn

Pierre lit un livre

u.lit(p, u): s/sn s\sn)/(s:.

xvxlivrexv

s:,.

xuplituxlivrex

s:,

xplitxlivrex

Curry-Howard

deduction / or \ - elimination / or \ - introduction hypothesis discharged

hypothesis normalisation

-term application abstraction variable bound variable -reduction

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

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/AB

BA

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/AB

BA

B

[A]

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/AB

[A]

BA

A

B

Normalisation and -reduction

B/AB

[A]

BA

A

B

(xA.’B A) ’B[A/xA]

sequent calculus

(intuitionist) sequent

BAAAA ni ,...,,...,, 21

antecedent consequent

To prove : C ,,, A/B

amounts to prove :

B

and then :

A C ,,

Lambek calculus(with product)

(sequents)

AA

CBACBA

,\,,,,

CABCBA

,,/,,,

BAAB/

,

ABAB\

,

CBACBA

,,,,,

BABA

,

CCAA

,,,,

A fundamental restriction:non empty antecedents

• a simple exercise

• a very simple exercise

• *a very exercise

npnnnnnp

npnnnpnpnpnn

nn

,/,/

,/

npnnnpnpnnnnnp

n

,nnnnnn /),//()/(,/,/,/

nnn ...nnnn //

,/

nn

,,,

npnnnpnpnnnnnp

nnnn ),//()/(///...

nn /

What sequent calculus reveals to us…

• cf. classical logic (some rules)

• (note the symmetries)

BA

BA

,

,,

',,,',

',',

BA

BA

',,',

',',

BA

BA

,

,,

BA

BA

',,',

',',

BA

BA

,

,,

BA

BA

• but also: (on the two sides)

• + axiom and cut-rule

',,,

',,,

AB

BA

',,

',,,

A

AA

A,

Permutation

Contraction

Weakening

Lambek calculus = intuitionistic logic WITHOUT A, C, P

Intuitionistic multiplicative linear logic

(+ restriction on non empty antecedents)

AA

subformula property

CC

,,,,,

CC

,,,,,

,

,

AA B BB/A

A/B

A\B

B\AAB AB

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

snsnssnsnsnsnnnnsn /)\()//()\(/

le livre que Pierre lit

snsnssnsnsn)//(sn)\(nnnsn /)\(/

snsnssnsnnnsn /)\(/ sn)//(sn)\(n

sn\nnnnsnsn/ssnssnsn //)\(

snsnssnsnnnsn /)\(/ sn)//(sn)\(n

sn\nnnnsnsnssnsn //)\( sn/sssnsnssnsn /)\(

snsnssnsnnnsn /)\(/ sn)//(sn)\(n

sn\nnnnsnsnssnsn //)\( sn/sssnsn sns)/\(snsssnsnsnsn \

snsnssnsnnnsn /)\(/ sn)//(sn)\(n

sn\nnnnsnsnssnsn //)\( sn/sssnsn sns)/\(snssnsnsn s\sn

sssnsn

snsnssnsnnnsn /)\(/ sn)//(sn)\(n

snnnsnsnssnsn n\nsn/s //)\(ssnsn sns)/\(snssnsnsn s\sn

sssnsn

snnnsnnn /snsnnn

But…cut-rule

CC

,,,,A A

L {Cut} = L

Fortunately : Cut-elimination theorem

Labelled Lambek calculus

AxAx ::

CBAC BA

:,\:,,:,,:

:

BAAB/:::,

f f()

x ux.u

Pierre lit Tintin(Pierre reads Tintin)

ssnsnssnxyyxsn ::/)\(:),(..: ?tlitp

sssnxyyxsnsnsn :\:)(),(..::: ?tlitptt

Pierre lit Tintin

ssnsnssnxyyxsn ::/)\(:),(..: ?tlitp

sssnyysnsnsn :\:),(.::: ?tlitptt

ssyysnsn ::))(,(.:: ?ptlitpp

Pierre lit Tintin

ssnsnssnxyyxsn ::/)\(:),(..: ?tlitp

sssnyysnsnsn :\:),(.::: ?tlitptt

sssnsn ::),(:: ?tplitpp

Pierre lit Tintin

ssnsnssnxyyxsn ::/)\(:),(..: ?tlitp

sssnyysnsnsn :\:),(.::: ?tlitptt

sssnsn :),(:),(:: tplittplitpp

Pierre lit Tintin

ssnsnssnxyyxsn ::/)\(:),(..: ?tlitp

sssnyysnsnsn :\:),(.::: t)lit(p,tlitptt

sssnsn :),(:),(:: tplittplitpp

Pierre lit Tintin

ssnsnssnxyyxsn ::/)\(:),(..: t)lit(p,tlitp

sssnyysnsnsn :\:),(.::: t)lit(p,tlitptt

sssnsn :),(:),(:: tplittplitpp

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

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!

– parasitic gaps• The book John filed _ without reading _

(linearity properties)

– empty signs• The book John read

(cf. non empty antecedents)

Extensions

• Multimodal Categorial Grammar (Moortgat, Oehrle, Morrill and their students)– ref. Categorial Type Logics in

• Van Benthem and ter Meulen (HLL)

see further…

A « cousin »

• Minimalist Grammars:– Inspired by Chomsky’s minimalist program– Ed. Stabler

• they have also type-logical formulations:– W. Vermaat– Retoré – Lecomte

see further…

or another day…

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)