Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
LEXICAL DEFINITIONS OF PREPOSITIONS
1
Locative prepositions
1.1
Preliminaries
(1)
A locative PP denotes a sets of vectors located in the reference object (a region)
a.
outside the barn: pointing to a point external to the barn
b.
near the barn: having a short length
c.
between the barn and the house: pointing to a point in the convex hull of the barn and the
house
d.
(2)
above the barn: pointing in the direction of the upward axis
Some questions
a.
What do we mean by ‘located in the reference object’?
b.
How do we define ‘external’ (and ‘internal’)?
c.
What role does vector ‘length’ play?
d.
What is the ‘convex hull’ of two objects?
e.
What are ‘axes’ and what role do they play?
1.2
About eigenspaces, boundaries, and closest vectors
(3)
Recall: The location function loc maps every (physical) object to its eigenspace, its location in
space.
(4)
Eigenspaces are topologically simple
a.
They are convex (if the points p and q are in the eigenspace, then so is every point
between p and q)
b.
They are closed (the eigenspace includes its boundary)
c.
They are non-trivial (non-empty and properly contained in Dp)
v4
v3
v1
P
Q
v2
v2
Figure 1
Figure 2
1
v3
v1
Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
(5)
By ‘located in the reference object’ we mean: having its starting point in the topological
boundary of the eigenspace of the reference object (like v3 and v4 in Figure 1)
(6)
DEFINITION 1 (Boundary vectors): Let v Î Dv be a vector and A Í Dp a set of points. We call
v a boundary vector of A, and denote boundary(v,A) iff s-point(v) is in b(A), the boundary of A.
(7)
(8)
(9)
a.
three kilometers outside town
b.
deep inside the cave
a.
P is diagonally above the box (v1 but not v2 in Figure 2)
b.
Q is exactly three inches outside the box (v3 but not v2 in Figure 2)
By ‘located in the reference object’ we mean: being a closest vector that has its starting point in
the topological boundary of the eigenspace of the reference object (like v1 and v3, but not v2 in
Figure 2)
(10)
DEFINITION 2 (Closest vectors). Let v Î Dv be a boundary vector of a set of points A Í Dp.
We say that v is a closest vector to A and denote closest(v,A) iff for every vector w Î Dv that is a
boundary vector of A s.t. e-point(v) = e-point(w): |v| £ |w|.
1.3
Inside and outside
(11)
DEFINITION 3 (Internally/externally closest vectors). Let v Î Dv be a closest boundary vector
of a set of points A Í Dp. In case e-point(v) Î A we call v internally closest to A and denote
int(v,A). Otherwise, we call v externally closest to A and denote ext(v,A).
(12)
(13)
in, inside:
inside¢ = def lA.lv.int(v,A)
outside:
outside¢ =def lA.lv.ext(v,A)
a is inside b iff loc(a) Í loc(b)
a is outside b iff loc(a) Ç loc(b) = Æ
(14)
(15)
1.4
a.
in/inside the suitcase
b.
in/?inside the tree
c.
?in/inside the wall
a.
the lemon in the bowl
b.
the coffee in the cup
Distance prepositions
2
Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
(16)
The norm (or length) |v| of a vector v is defined as Öf(v,v), where f is the positive scalar product.
(17)
on, at: on¢ = at¢ = def lA.lv.ext(v,A) Ù |v| < r0
near¢ = def lA.lv.ext(v,A) Ù |v| < r1
near:
where r0 and r1 two small positive numbers r0 » 0 and r0 << r1.
(18)
(19)
a.
The particle is near the nucleus
b.
The rocket is near the planet
a.
the book on the shelf
b.
the lamp on the ceiling
1.5
Between
(20)
between ... and ...:
between¢ =def lA.lB.lv. [ext(v,A) Ú ext(v,B)] Ù e-point(v) Î co(A È B)\ A\ B
B
l
A
A
yB
D
C
Figure 3
(21)
Figure 4
For v, w in a space V, the line segment [v,w] is the set {sv + (1 - s)w: 0 £ s £ 1 }. A set A Í V is
convex iff for all v, w Î A: [v,w] Í A. The convex hull of A, denoted co(A), is the smallest
convex subset of V containing A.
(22)
Transitivity
A is between B and C
D is between A and B
D is between B and C
3
Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
(23)
a.
The tree is between/among/amid the houses
b.
Duisberg lies between Essen and Düsseldorf (Habel 1989)
1.6
Projective prepositions
(24)
a.
The axes are represented by three free orthogonal unit vectors in V, up, right, front
b.
c(a,v) is the component of v along the axis a, a scalar
c.
v^a means the projection of v on the axis orthogonal to a
up(w)
vup
w
v
v^up
Figure 5
(25)
above: above¢ =def lA.lv.ext(v,A) Ù c(up,v) > |v^up|
below, under: below¢ =def lA.lv.ext(v,A) Ù c(-up,v) > |v^-up|
in front of: in_front¢ =def lA.lv.ext(v,A) Ù c(front,v) > |v^front|
behind: behind¢ =def lA.lv.ext(v,A) Ù c(-front,v) > |v^-front|
beside: beside¢ =def lA.lv.ext(v,A) Ù |c(right,v)| > |v^right|
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Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
Q(0.25)
Q(0.5)
Q(0)
Q(1)
Q(0.75)
Figure 7
Figure 6
(26)
(27)
(28)
Absolute (environmental) axes
a.
The body is under the car
b.
north of, upstream of, downwind of
Intrinsic axes
a.
Otto has a scar above his left eye
b.
Don’t stand behind that horse!
c.
My mouse is always to the right of my keyboard
Relative (deictic) axes
a.
Bob is standing in front of the tree (from my point of view)
b.
I can see him standing to the left of the tree
2
Directional prepositions
(29)
a.
The car will go into/through the garage Þ The car will be in the garage
b.
towards ~ away from
c.
The car drove around the garage
d.
The trees are standing along the river, This road leads to the city, John looked through
the window
(30)
a.
A directional preposition maps a reference object to a set of paths
b.
A path is a sequence of vectors, here defined as a function Q from the real [0,1] interval
to vectors
at
on
in
source
from
off
out of
goal
to
onto
into
route
via
across
through
5
above
beside
over
along
Introduction to Vector Space Semantics - ESSLLI 2001 Helsinki
Table 1
(31)
DEFINITION 4 (Closest path). We say that a path Q from [0,1] to Dp is the closest path to a set
of points A Í Dp and denote closest(Q,A) iff for every x Î [0,1]: Q(x) is a closest vector to A.
(32)
source:
dir0(P) = lQ.closest(Q,A) Ù P(A)(Q(0))
goal:
dir1(P) = lQ.closest(Q,A) Ù P(A)(Q(1))
route:
dir$(P) = lQ.closest(Q,A) Ù $x Î [0,1] [ P(A)(Q(x)) ]
at
source
on
from’ =
0
goal
route
in
off’ =
0
above
beside
out_of’ =
dir (at’)
dir (on’)
dir0(in’)
to’ = dir1(at’)
onto’ =
into’ =
dir1(on’)
dir1(in’)
via’ =
across’ =
through’ =
over’ =
along’ =
dir$(at’)
dir$(on’)
dir$(in’)
dir$(above’)
dir$(beside’)
Table 2
(33)
a.
The duck came from under the bridge
(dir0(under’))
b.
The duck went under the bridge
(dir1(under’))
a.
The duck passed under the bridge
(dir$(under’))
(34)
Not definable in terms of locative prepositions: towards, away from, around
(35)
Reverse prepositions:
~P =def lA.lQ.$Q Î P(A)[ Q = ~Q]
where ~Q =def lx.Q(1 - x )
6