# THE NATURAL LANGUAGE PROCESSING

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Oct 24, 2013 (4 years and 6 months ago)

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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

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

Combinatory Categorial Grammar

Two concepts :

Syntactic category

Semantic category

Example :

the category of

is

\
NP : np1)/NP : np2

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)

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) :
]

4

[S : ((
B

(
C*

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

5

[S : ((
C*

John
) (
loves

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

(
B
)

6

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

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

(
>dec
)

7

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

John
)]
-
[S
\
N : (

(
loves Mary
))]

(
<
)

8

[S : ((
C*

John
) (

(
loves Mary
)))]

(
>
)

9

[S : ((
C*

John
) (

(
loves Mary
)))]

10

[S : ((

(
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

]

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