THE NATURAL LANGUAGE PROCESSING

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CATEGORIAL GRAMMARS AND

THE NATURAL LANGUAGE PROCESSING

Ismaïl Biskri

Mathematics and Computer
-
Science Department

University of Quebec in Trois
-
Rivières


www.uqtr.ca/~biskri

biskri@uqtr.ca







HUSSERL

(1913)


Philosophical Origins.


Notions of :


Categorem


Syncategorem


Example :


Noun : Categorem


Sentence : Categorem


Verb : Syncategorem

LESNEWSKI (1922)


Logical foundations.


Two kind of expressions :


Noun


Proposition



Noun : objects, class of objects.


Proposition : statement (describing a
“state”).





LESNEWSKI (1922)


Nouns and Propositions are Categorems.


Other expressions are Syncategorems.




Syncategorem acts like an operator.


Categorem acts like an operand.

LESNEWSKI (1922)



Inferential System


We

assume

that

we

have

a

set

of

basic

types


The

set

of

all

types

is

defined

recursively

as

follows
:


Basic

types

are

types

;


If

x

and

y

are

types

then

Fxy

is

a

type
.



(F

is

an

applicative

operator

;

F

is

applied

to

an

expression

of

type

x,

it

yields

an

other

expression

of

type

y)


AJDUKIEWICZ (1935)


Basic expressions (categories) :


Noun (N)


Sentences (S)


If x and y are categories then is a
category


Reduction rules



y


x



y


x

y

x

y

x

y

x

AJDUKIEWICZ (1935)

Example


John




laughs

------




--------

N








-----------------------------------

S

N

S

BAR
-
HILLEL (1953)


Basic expressions (categories) :


Noun (N)


Sentences (S)


If x and y are categories then x/y and x
\
y are
categories


Reduction rules :


x/y y


x



y y
\
x


x

BAR
-
HILLEL (1953)

Example


John


admires


Mary

------


----------


-------

N



(N
\
S)/N


N




-----------------------




N
\
S

--------------------

S

LAMBEK (1958, 1962)


Lambek Calculus.


We will use Steedman’s notation


X/Y will be X/Y


Y
\
X will be X
\
Y


Many axioms


Many inference rules


Many theorems


LAMBEK (1958, 1962)

Axioms



X






X


(reflexivity)



(X


Y)


Z



X


(Y


Z)

(associativity)



X


(Y


Z)



(X


Y)


Z

(associativity)

LAMBEK (1958, 1962)

Inference rules


If X


Y and Y


Z

then X


Z

(transitivity)


If X


Y


Z


then X


Z/Y


If X


Y


Z


then Y


Z
\
X


If X


Z/Y


then X


Y


Z


If Y


Z
\
X


then X


Y


Z



LAMBEK (1958, 1962)

Some Theorems


X






(X


Y)/Y


(Z/Y)


Y




Z


Y






Z
\
(Z/Y)


(Z/Y)


(Y/X)




Z/X


Z/Y





(Z/X)/(Y/X)


(Y
\
X)/Z





(Y/Z)
\
X

ADES, STEEDMAN (1982)


Combinatory Categorial Grammar


Two concepts :


Syntactic category


Semantic category


Example :

the category of
admires

is


(S : admire' np2 np1
\
NP : np1)/NP : np2



ADES, STEEDMAN (1982)

Some rules


Functional application (>) :


X/Y : f


Y : y



X : f y



Functional composition (>B) :


X/Y : f


Y/Z : g



X/Z :

z(f(gz)



Type Raising (>T) :


X : x





Y/(Y
\
X) :

f(fx)


ADES, STEEDMAN (1982, 1989))

Example


John
-


loves
-





Mary

------



-----------




-------

N
:

John‘

(S
:
loves‘

np
2

np
1
\
NP
:

np
1
)/NP
:

np
2

NP
:

Mary'

---------
>T

S

:

pred

John'/(S
:

pred

John'
\
NP
:

John')

-----------------------------------------------------------
>B

S
:

loves'

np
2

John'/NP
:

np
2

-------------------------------------------------------------------------
>

S
:

loves'

Mary'

John'


BISKRI , DESCLES (1995, 1997)


Applicative Combinatory Categorial Grammar.



Canonical association between Combinatory
Categorial rules and Combinators of Combinatory
Logic (Curry, Feys, 1958).



Combinatory Categorial rules : syntactic parsing.



Combinatory Logic : functional semantic parsing

BISKRI, DESCLES (1995, 1997)

Combinatory Logic


Combinators :
B
,
C
,
C
*
,
S
, etc.



Beta
-
Reduction rules :

B

f g x


f (g x)


;
C
*

x f


f x




Combinatory expression Normal Form


B C
*

x y z t


is not in normal form


B

C
*

x



is in normal form


x (y z)



is in normal form





BISKRI, DESCLES (1995, 1997)

Some rules


Functional application (>) :


X/Y : f


Y : y




X : f y



Functional composition (>B) :


X/Y : f


Y/Z : g



X/Z :
B

f g



Type Raising (>T) :


X : x





Y/(Y
\
X) :
C
*

x


BISKRI, DESCLES (1995, 1997)

Example 1

1
[N:
John
]
-
[(S
\
N)/N:
loves
]
-
[N:
Mary
]

Typed concatenated structure


2

[S/(S
\
N):(
C*

John
)]
-
[(S
\
N)/N:
loves
]
-
[N:
Mary
]

(
>T
)

3
[S/N:(
B

(
C*

John
)
loves
)]
-
[N:
Mary
]


(
>B
)

4
[S:((
B

(
C*

John
)
loves
)
Mary
)]



(
>
)


Typed applicative structure

5

[S : ((
B

(
C*

John
)
loves
)
Mary
)]



6

[S : ((
C*

John
) (
loves Mary
))]



(
B
)

7

[S : ((
loves Mary
)
John
)]




(
C
*
)


BISKRI, DELISLE (2000)

Example 2 :


[N/N
:
la
]
-
[N
:
liberté
]
-
[(S
\
N)/N
:
renforce
]
-
[N/N
:
la
]
-
[N
:
démocratie
]

2
.


[N
:
(
la
-
liberté
)]
-
[(S
\
N)/N
:
renforce
]
-
[N/N
:
la
]
-
[N
:
démocratie
]


(>)

3
.


[S/(S
\
N)
:
(
C
*

(
la

liberté
))]
-
[(S
\
N)/N
:

renforce
]
-
[N/N
:

la
]
-
[N
:

démocratie
]

(>T)

4
.


[S/N

:

(
B

(
C
*

(
la

liberté))

renforce)
]
-
[N/N

:

la
]
-
[N

:

démocratie
]

(>B)

5
.


[S/N

:

(
B

(
B

(
C
*

(
la

liberté))

renforce)

la)
]
-
[N

:

démocratie
]


(>B)

6.
[S

:

((
B

(
B

(
C
*

(
la

liberté))

renforce)

la)

démocratie
)]



(>)


7
.


[S

:

((
B

(
B

(
C
*

(
la

liberté))

renforce)

la)

démocratie
)]

8
.


[S

:

((
B

(
C
*

(
la

liberté))

renforce)

(
la

démocratie
))]



B

9
.


[S

:

((
C
*

(
la

liberté))

(
renforce

(
la

démocratie
)))]



B

10
.


[S

:

((
renforce

(
la

démocratie
))

(
la

liberté
)))]




C*

11
.


[S

:

renforce

(
la

démocratie
)

(
la

liberté
)]


BISKRI, DELISLE (2000)

Example 3


1
.


[(S/N
1
)/N
2
:
thoudaiimou
]
-
[N
1
:
elhouriyathou
]
-
[N
2
:
e
ddimouqratiyatha
]

2
.


[(S/N
1
)/N
2
:
thoudaiimou
]
-
[S
\
(S/N
1
)
:
(
C*
elhouriyathou
)]
-
[N
2
:
eddimouqratiyatha
]





(<T)

3
.


[S/N
2

:

(
B

(
C
*

elhouriyathou
)

thoudaiimou)
]
-
[N
2

:

eddimouqratiyatha
]








(<Bx)

4
.


[S

:

((
B

(
C
*

elhouriyathou)

thoudaiimou)

eddimouqratiyatha
)]

(>)

5
.



[S

:

((
B

(
C
*

elhouriyathou)

thoudaiimou)

eddimouqratiyatha
)]

6
.


[S

:

((
C
*

elhouriyathou)

(
thoudaiimou

eddimouqratiyatha
))]

B

7
.


[S

:

((
thoudaiimou

eddimouqratiyatha
)

elhouriyathou
)]


C
*

8
.


[S

:

thoudaiimou

eddimouqratiyatha

elhouriyathou
]


BISKRI, DESCLES (1995)

The Backward Modifier : Example 4


1

[N :
John
]
-
[(S
\
N)/N :
loves
]
-
[N :
Mary
]
-
[(S
\
N)
\
(S
\
N) :
madly
]




4

[S : ((
B

(
C*

John
)
loves
)
Mary
)]
-
[(S
\
N)
\
(S
\
N) :
madly
]

5

[S : ((
C*

John
) (
loves

Mary
))]
-
[(S
\
N)
\
(S
\
N) :
madly
]

(
B
)

6

[S/(S
\
N) : (
C*

John
)]
-
[S
\
N : (
loves Mary
)]
-
[(S
\
N)
\
(S
\
N) :
madly
]








(
>dec
)

7

[S/(S
\
N) : (
C*

John
)]
-
[S
\
N : (
madly

(
loves Mary
))]

(
<
)

8

[S : ((
C*

John
) (
madly

(
loves Mary
)))]



(
>
)


9

[S : ((
C*

John
) (
madly

(
loves Mary
)))]


10

[S : ((
madly

(
loves Mary
))
John
)]



(
C*
)



BISKRI, DESCLES (1995)

Coordination

a)

Two

segments

of

the

same

kind,

with

the

same

structure

and

contiguous

to

AND

:



[
John

loves
]
S/N

and

[
William

hates
]
S/N

these

pictures

b)

Two

segments

into

an

elliptic

construction

:



John

loves

[
Mary

madly
]

and

[
Jenny

wildly
]



[
John
]

loves

[
Mary
]

and

[
William

Jenny
]

c)

Two

segments

of

different

structures

:



Mary

walks

[
slowly
]

and

[
with

happiness
]
.



John

[
sings
]

and

[
plays

the

violin
]
.

d)

Two

segments

without

distributivity

:



The

flag

is

[
white
]

and

[
red
]




(≠
The flag is white and the flag is red
).


BISKRI, DESCLES (1995)

Example 5

1

[N:
John
]
-
[(S
\
N)/N:
loves
]
-
[N:
Mary
]
-
[CONJD:
and
]
-
[(S
\
N)/N:
hates
]
-
[N:
Jenny
]


...

4

[S:((
B

(
C*

John
)
loves
)
Mary
)]
-
[CONJD:
and
]
-
[(S
\
N)/N:
hates
]
-
[N:
Jenny
]

5

[S:((
B

(
C*

John
)
loves
)
Mary
)]
-
[CONJD:
and
]
-
[S
\
N:(
hates

Jenny
)] (
>
)

6

[S:((
C*

John
) (
loves

Mary
))]
-
[CONJD:
and
]
-
[S
\
N:(
hates

Jenny
)] (
B
)

7

[S/(S
\
N):(
C*

John
)]
-
[S
\
N:(
loves

Mary
)]
-
[CONJD:
and
]
-
[S
\
N:(
hates

Jenny
)]








(
>dec
)

8

[S/(S
\
N):(
C*

John
)]
-
[S
\
N:(
F

and

(
loves

Mary
) (
hates

Jenny
))] (
<CONJD>
)

9
[S:((
C*

John
) (
F

and

(
loves

Mary
) (
hates

Jenny
)))]



(
>
)


10

[S : ((
C*

John
) (
F

and

(
loves

Mary
) (
hates

Jenny
)))]

11

[S : ((
F

and

(
loves

Mary
) (
hates

Jenny
))
John
)]


(
C*
)

12

[S : (
and

((
loves

Mary
)
John
) ((
hates

Jenny
)
John
))]


(
F
F