But… Why not to have a syntax built on the same principles as those of semantic composition?
-
Upload
amia-maloney -
Category
Documents
-
view
214 -
download
1
Transcript of But… Why not to have a syntax built on the same principles as those of semantic composition?
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)