But

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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
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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
• Marie dort s
• np s

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• Start marie np, marie dort s
• Marie dort s
• np np\s s

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• dort profondément np\s
• np\s np\s

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

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

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• dort profondément np\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
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B hypothesis labelled ni
the hypothesis ni is discharged
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Example type-raising 
A\B1
A
B/(A\B)
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Le livre que Pierre lit
sn1
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but

• natural deductions are precisely ?-terms !

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aB
f A/B
f(a)A
xBi
G
uA
i
lx.uA/B
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Pierre lit Tintin
(sn\s)/sn
sn
lx.ly.lit(y,x)
psn
t
(ly.lit(y,t))(p) s
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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)
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Pierre lit un livre
lu.lit(p, u) s/sn
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Curry-Howard

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

• l-term
• application
• abstraction
• variable
• bound variable
• b-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
?
A
B/A
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
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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
A
B/A
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

B/A
B
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Normalisation and ?-reduction
?B?A/xA
(?xA.?B ?A)
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sequent calculus
(intuitionist) sequent
consequent
antecedent
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A/B
Q
To prove
amounts to prove
Q
B
and then
A
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Lambek calculus(with product)(sequents)

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A fundamental restrictionnon empty antecedents

• a simple exercise
• a very simple exercise
• a very exercise

n
np
n
n
n
n
np
,
/
,
/
a
np
n
n
np
,
,
/
a
n
n
...
a
,
,
,
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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
Weakening
Contraction
37

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

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subformula property

A
A
B
B
B/A
A\B
B
A
A
B
A/B
B\A
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le livre que Pierre lit(the book that Peter
reads)
le
livre
que
Pierre
lit
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Butcut-rule

A
A
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Fortunately Cut-elimination theorem

Cut L
L
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Labelled Lambek calculus

a
f(a)
f
x
u
lx.u
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Pierre lit Tintin(Pierre reads Tintin)
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Pierre lit Tintin
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Pierre lit Tintin
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Pierre lit Tintin
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Pierre lit 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)