19 April 2022

AperTO - Archivio Istituzionale Open Access dell'Università di Torino

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Constructive Adpositional Grammars

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Table of Contents

List of Figures

! !

List of Tables

Legenda

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2

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$

S

!!!!!

!

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1

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(& C ,S)

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!

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& C

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$

(& C ,S)

!!!!!

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!

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= E !& D

= E !% A

$

(= E ,S)

!!!!!

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!

!

& D

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$

(= E ,S)

!!!

!!

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!!

!

!

% A

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(% A , S)

& D = E = E % A

! "illoc

& D % A & D & D

st st st "loc "illoc

! % A & D

! ! & D & D

% A & D

! ! % A % A

! % A % A

! % A & C

! & D & D

! % A & D

! & C % A

! ! = E % A

! ! % A & D

! ! % A % A

! = E = E

% A & D

& C & D

& D % A = E & D = E

ChapterSeven

Conclusion

AppendixA

Examples

!"I2

2

()O2 !!

I21

( )O1 ()I2

!!I2

2

( )O2 !"

I21

( )O1 ()I2

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1

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1

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1

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2

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2

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I21

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3

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1

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noun: proper, singular verb: [tense: simplepresent: [3rd person, singular]]

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X noun singular !!I1 I 3rd

person singular I1

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2

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aux

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I22

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1

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aux

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!Yact !"

I22

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1

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aux

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I22

&&Xact I2 !Yact

!!I2

1

&&Xact I2

aux

"I2

2

( )Y !"I2

1

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"E > I

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I22

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2 > A($$$$ )Yact

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1

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mean !Yaux

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1

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O "O!D I

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mean !Yaux

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I21

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I21

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1

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be

!!I2

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()I2

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I21

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O'Xwalk

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()I2

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Z !!I3

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1

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E !!I1 X ( )I1 !G

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2

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AE A

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!!I2

2

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X I2

"I2+1

3

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2

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1

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"I2+1(1

3

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2

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1

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!I4+15

( )W "I4+1

4

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( )Y

!"I4+1

2

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2

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2

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1

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3

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2

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I22

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S % A ()(% A , S)

!!& D!!

I22

( )(Z,R) !!I2

1

R & D ()(& D,S)

!!& C!!

I22

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S ( )& C ()(& C ,S)

!!= E

!E( )(Y,S) !

!I2

2

( )R ( )(Y,S)

()(= E ,S)

!"( )' D (Declare)' D

!% A!"

Y"

Y()X = S > A ()Y ()Ix

!"% A()X = S ()% A

!% A( )& D (Itell)% A

!!& D!"

Ix ()Z !#Ix ()Y = R ()Ix

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!!& D( )(Z,R) ( )(& D,S)

!!& C!"

Ix ()Z !#Ix ()Y = S ()Ix

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!Ix ()R

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!"= E

()X = S ()= E

!' D!!

Ix S !"Ix Y = R Ix

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!!' D!"

I22

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!OA !

!OA O

!!I2

1

R I2 !"' DS ' D

!% A( )% A ()% A

!& D( )& D ()& D

!& C( )& C ()& C

!= E

( )= E ()= E

!"& D( )(& D,S) ( )(& C ,S)

!"& C( )(& C ,S) ( )(& D,S)

!& D( )(= E ,S) (tellme )(& D,S)

!% A( )(= E ,S) ( )(% A ,S)

AppendixB

The FormalModel

C G C G ObjC C

G = {Gi}i-GC !

i-GC Gi = ObjC i, j - GC i " j Gi . G j = / Gi " /

C G =

{Gi}i-GC W - G CG W 0Y1 1 · · · 1 Yn, (z1, . . . , zn)2 n 1 i 1 . . . n Yi - G \ {W} zi - W

C C i W

C G ={Gi}i-GC W - G &P,x P x

"

f j

#

j-P3

P = 0P3, a2 C GW

x C

V - G \ {W} j - P3 f j - HomC(x, y) y - V

V cod &P,x dom &P,x = x P &P,x & j,x f j

$

&P,x

%

x-U P x U

&P,U P U cod &P,U dom &P,U = U P & j,x j x &P,U

& P = 0Y1 1 · · · 1 Yn, (z1, . . . , zn)2 x x Y1, . . . , Yn z1, . . . , zn P &

& &

C G ={Gi}i-GC W - G

& =&

"

f j : x " c(x, j)#

j-P3

'

x-V

P = 0P3 = P1 1 · · · 1 Pn, (a1, . . . , an)2 V ' Q U & k 1 . . . n U = Pk

Q = 0P1 1 · · · 1 Pk(1 1 V 1 Pk+1 1 · · · Pn, (ak, a2 . . . , ak(1, a1, ak+1 . . . , an)2

(x, j1, . . . , jn) - C 1 P3 cod &( j1,..., jn),x = cod '( j1,..., jk(1,x, jk+1,..., jn), jk

C G = {Gi}i-GC W - G

& =&

"

f(y1,...,yn) : x " c(x, y1, . . . , yn)#

y1-P1,...,yn-Pn

'

x-V

P = 0P1 1 · · · 1 Pn, (a1, . . . , an)2 V

' ="

$

g(z1,...zm) : x " d(x, z1, . . . , zm)%

z1-Q1,...,zm-Qm

#

x-U

Q = 0Q1 1 · · · 1 Qm, (b1, . . . , bm)2 U ' + & dom ' = U = cod &

' + & =&

"

g(z1,...zm) + f(y1,...yn)

#

y1-P1,...,yn-Pn,z1-Q1,...,zm-Qm

'

x-V

dom ' + & = dom & = V cod ' + & = cod ' ' + &

0P1 1 · · · Pn 1 Q1 1 · · · 1 Qm, (a1, . . . , an, b1, . . . , bm)2 .

C G = {Gi}i-GC W - G C C G W G 0C,G,W,C2

M C x - ObjC f C dom f - M

cod f = x x - M f = idx

f : a " b C &P,V - C a - V f - &P,a &P,a a &P,V

& - C '1, . . . , 'n C 1

C

A A = 0C,G,U,C2 G

GC = {A, E,O,U} 4"

Ip : p C,G,U#

.

Ip

0C,G = {Gi}i-GC ,W,C2 & 0Y1 1 · · · 1 Yn, (a1, . . . , an)2 x

& ="

f(y1,...,yn) : x " c(x,y1,...,yn)

#

y1-Y1,...,yn-Yn

.

&

xf(y1 ,...,yn)

!! an(yn, an(1(. . . , a1(y1, x) . . .)) = c(x, y1, . . . , yn) - cod & .

x (y1, . . . , yn) &, yk (y1, . . . , yk(1, x, yk+1, . . . , yn)

an(yn, an(1(. . . , ak1(yk(1, a1(x, ak+1(. . . (ak(y1, yk)) . . .)) . . .))

c(x, y1, . . . , yn)

& &

0C,G = {Gi}i-GC ,W,C2

& =&

"

f(x,y1,...,yn) : x " c(x,y1,...,yn)

#

y1-Y1,...,yn-Yn

'

x-V

0Y1 1 · · ·1 Yn, (a1, . . . , an)2 V (y1, . . . , yn) - Y1 1 · · · 1 Yn

'(y1,...,yn) =

"

f(x,y1,...,yn)

#

x-V.

GC '

'

!(L,N) N L

L

a(x, T0) & a(T0, x) &, T1 5 T2 T1 = T [a(x,T0)] T2 = T [a(T0, x)] T1 T2 T1 T2 T1 T2 6

5 6

A A x y x 6 y y

T L

T T

T L = T T L T

T = a(T2,T1)

L1 = bk(tk, bk(1(. . . , b1(t1, t0) . . .))

L2 T1 6 L1 T2 6 L2 T1 L1 T2 L2 T 6 a(L2, L1)

T 6 a(L2, bk(tk, bk(1(. . . , b1(t1, t0) . . .)) 6

6 b1(t0, bk(tk, bk(1(. . . , a(t1, L2) . . .)) = L ,

L T

T [T1] T1 T1

6

U U = U 1 {!,",#}

! ! !A !E !O !

M M = M 7 M M M

C C

Ad(C) C 8 Ad(C) C C 8 C 8 Ad(C)

C Ad(C) Ad(C)

!

F T F(T ) !(a, n) a n !(a3, n) a3

F T F T = F T f F t f

F f = F + · · · + F t f

F t F t

R 8 R TR TR

M = 0M,GM ,WM ,CM2 L = 0L,GL,WL,CL2 ( : ObjM " ObjL L C = 0C,G,W,C2

C !M ObjL 7 ObjM M = {!M}

!M L M C L M (

C M !M

M G = GL 7 GM 7 M

C W = WL 7 WM 7 {!M}

C

C = CL 7 CM 7&&

"

' j : x " ((x)#

j-M

'

x-%: % - GM

'

.

C

C M L L ! ' M (

C L M C C M L C

L M

c C S C c f - S f + h f + h - S S c

S 3

f : b " c - S b - S 3 d

g : d " b d - S 3 S c B c S 3 c B

S c c S 3 f : c " b - S b - S 3 d b d3 - S 3 d3 d b c B

c B

c 0O, (!)2 v S c v v

I I

C J c C J(c) c

tc = { f : cod f = c} - J(c) c J(c)

S - J(c) h : d " c C

h,(S ) = {g : cod g = d h + g - S } - J(d) ;

S - J(c) R c h,(R) - J(d) h : d " c

S R - J(c)

0C, J2 J C

S - J(c) J h,(S ) h c h,(S )

J C 0C, J2 Sh(C, J) C Cop " Set 0C, J2 P C c C S - J(c) S " Hom((, c) Hom(S , P) # Hom(Hom((, c), P)

Sh(C, J)

C Sh(C, J)

Sh(C, J) C

G = 0V,!, S , P2 V ! S - V P A " % A - V % - (V 4 !), V ! S P

G = 0V,!, S , P2 G

L(G) =&

x - !, : SG

(" x'

,

G

(" # %, ) - (V 4 !), % # ) v - V % = %3 · v · %33

) = %3 · )3 · %33 (v " )3) - P

L G L = L(G)

G = 0V,!, S , P2 G

("9

V 1 (V 4!), C3 * V 4 ! S

G

(" *

G

("

C C3 7 1 7 1 !

G ObjC {c} c G ObjC C G

(" G = 0C,G, {!},C2 {S , !}

L(G) G P

L(G) S

L(G) G c - !, S

" L(G) G = 0V,!, S , P2

G = 0V,!, S , P2 G3 = 0V 3,!, S , P32 L(G) =L(G3) G3 p3 A " B ·C B,C - V 3 A " a a - !

C, !,

(A " a) - P3 a - GA

(A " B · D) - P3 b - GB d - GD b · d - GA b " b · d d " b · d C, 0GD, (!)2 GB 0GB, (!)2 GD C

C C, (!

v-V 3 Gv

)

4 {!}

*

C, {!} 4+

v-V 3

Gv, {!},C

,

L(G) = L(G3) = GS GS

Notes

" "

gov gov dep adp gc

tr lm tr/lm tr lm

VP

Y X Y Z W Sold Objective Y

Eat A-Eat H-Eat

I > A > Y

$ ! !

I

Zmean

Price

1 3

"

& ' &, '

1 1 G

"

"

Bibliography

Index

" # # A + B + C C A B

#

# +

# +

# #

# #

# #

# +

# ##

##

#

# + +

##

# +

#

# + #

# + #

# + #

# + +

#

# +

#

# + #

# +

#

# + ## "

" ## "

##

## +

## +

## + +

# + +

# + +

## +

## + +

## + +

## + +

#

# + #

# +

#

# +

# +

# +

#

# #

# +

## + +

## +

# +

# + ## +

## +

## +

##

##

##

## +

##

## +

#

# + +

## + +

#

# +

#

# +

# + #

# +

## +

# + +

# +

#

# + # +

# +

#

#

# + #

# +

#

# + #

# + ! #

# + ! #

# + #

# + #

#

# #

# +

# +

# +

#

# +

#

#

# + #

# + #

# + +

# ## +

# # + # # + # # +

## +

## +

## +

# ## +

## +

## +

##

## +

# + +

## +

## +

## +

## +

#

# + +

# + +

# +

## +

## + +

##

##

## +

## + +

## +

##

## +

## + +

# +

# +

##

##

#

# + #

# +

#

# +

## +

#

# #

# +

#

# +

# + #

# + #

# +

#

# +

## +

" ## + +

## + +

#

# +

#

# + #

# + +

#

# #

# + ##

#

# #

# + #

# #

# + #

# +

## +

##

##

##

##

## + +

#

# + #

# + +

#

# +

#

# + + #

# + #

# + #

# + #

# + #

# #

# + +

#

# +

# +

#

# + #

# + $ #

# + #

# +

#

# + #

# + #

# + #

# +

#

# +

## + +

##

## +

#

# +

# #

## +

# +

## + +

## + +

## + +

## +

##

#

# + # #

# + #

# + #

# + +

#

# +

## +

## +

#

#

#

# +

#

# +

##

##

# +

# +

# +

# +

#

# +

#

# + #

# +

#

#