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Delim
ited
con
tinu
atio
ns
inn
atu
ral
lan
gu
age:
Qu
an
tifica
tion
an
dpola
rityse
nsitiv
ity
Ch
un
g-ch
ieh
Sh
an
Harv
ard
Un
iversity
Con
tinu
atio
ns
Work
shop
17
Jan
uary
20
04
Com
pu
tatio
nal
Lin
gu
istics
1
Wh
at
alin
gu
istca
res
abou
t
En
tailm
en
t
every
stud
en
ten
joyed
aco
nfe
ren
ce0
every
stud
en
ten
joyed
PO
PL
no
stud
en
ten
joyed
aco
nfe
ren
ce`
no
stud
en
ten
joyed
PO
PL
astu
den
ten
joyed
aco
nfe
ren
ce0
astu
den
ten
joyed
PO
PL
most
stud
en
tsen
joyed
aco
nfe
ren
ce0
most
stud
en
tsen
joyed
PO
PL
Am
big
uity
Did
som
estu
den
ten
joy
every
con
fere
nce
?
∃x.∀
y.en
joyed
(x,y
)
∀y.∃
x.en
joyed
(x,y
)
Did
an
ystu
den
ten
joy
every
con
fere
nce
?
Acce
pta
bility
*every
stud
en
ten
joyed
an
yco
nfe
ren
cen
ostu
den
ten
joyed
an
yco
nfe
ren
ce*a
stud
en
ten
joyed
an
yco
nfe
ren
ce
*m
ost
stud
en
tsen
joyed
an
yco
nfe
ren
ce
Th
ista
lkd
eals
with
En
glish
,bu
tth
eappro
ach
hopefu
llyexte
nd
sto
oth
er
lan
gu
ages
(wh
ichare
diffe
ren
t!).2
Tra
nsla
tion
toa
logica
lm
eta
lan
gu
age
Every
stud
en
ten
joyed
aco
nfe
ren
ce0
Every
stud
en
ten
joyed
PO
PL
∀x.stu
den
t(x)⇒
∃y.co
nf(y
)∧
enjo
yed(x
,y)
0∀x.stu
den
t(x)⇒
enjo
yed(x
,popl)
〈som
etru
thco
nd
ition
on
mod
els〉
0〈so
me
truth
con
ditio
non
mod
els〉3
Th
egu
idin
gan
alo
gy
Pro
gra
mm
ing
lan
gu
ages
Natu
ral
lan
gu
ages
desire
dbeh
avio
rsp
eaker
jud
gm
en
tsobse
rvatio
ns
at
gro
un
dty
pe
truth
con
ditio
ns,
etc.
type
syste
msy
nta
xd
en
ota
tion
al
sem
an
ticsd
en
ota
tion
al
sem
an
tics
com
pu
tatio
nal
side
effe
cts“lin
gu
isticsid
eeffe
cts”co
ntro
leffe
ctsqu
an
tifica
tion
,pola
rity,etc.
Com
pu
tatio
nal
side
effe
ctsin
the
logica
lm
eta
lan
gu
age
...
...han
dle
s“lin
guistic
side
effects”
Sta
tein
the
logica
lm
eta
lan
gu
age
......h
an
dle
spro
nou
ns
an
dbin
din
g
Con
trol
opera
tors
inth
elo
gica
lm
eta
lan
gu
age
......h
an
dle
squ
an
tifica
tion
an
dpola
ritysen
sitivity
4
Ou
tline
XO
verv
iew
IA
simple
gra
mm
atica
lfo
rmalism
•Q
uan
tifica
tion
with
shift
an
dre
set
•Q
uan
tifier
scope
am
big
uity
•Pola
rityse
nsitiv
ity
Com
pu
tatio
nal
side
effe
ctsin
the
logica
lm
eta
lan
gu
age
...
...han
dle
s“lin
guistic
side
effects”
Sta
tein
the
logica
lm
eta
lan
gu
age
......h
an
dle
spro
nou
ns
an
dbin
din
g
Con
trol
opera
tors
inth
elo
gica
lm
eta
lan
gu
age
......h
an
dle
squ
an
tifica
tion
an
dpola
ritysen
sitivity
5
Asim
ple
gra
mm
atica
lfo
rmalism
Alice
en
joyed
PO
PL.
*A
liceen
joyed
.*A
liceen
joyed
Bob
PO
PL.
JAliceK
=alice
:T
hin
g
JPO
PLK
=popl
:T
hin
g
Jen
joyedK
=en
joyed
:T
hin
g→́
(Thin
g→̀
Bool)
f′x
=f(x
):β
wh
ere
f:α→́
β,
x:α
x8f
=f(x
):β
wh
ere
f:α→̀
β,
x:α
Alice
en
joyed
PO
PL
alice
8(en
joyed
′popl)
:B
ool
Alice
PO
PL
en
joyed
alice
8(p
opl8en
joyed
):B
ool
Rig
ht-a
ssocia
tive
by
con
ven
tion
.
Nota
tion
al
varia
nt
of
com
bin
ato
ryca
tegoria
lgra
mm
ar.
6
Qu
an
tifica
tion
Every
stud
en
ten
joyed
PO
PL.
∀x.stu
den
t(x)⇒
enjo
yed(p
opl)(x
)
JEvery
stud
en
tK=
??E
very
stud
en
ten
joyed
PO
PL
studen
t:T
hin
g→
Bool
JeveryK
=λr.λ
s.∀x.r(x
)⇒
s(x):(T
hin
g→
Bool)→́
(Thin
g→̀
Bool)→́
Bool
Jsom
eK=
λr.λ
s.∃x.r(x
)∧
s(x)
:(T
hin
g→
Bool)→́
(Thin
g→̀
Bool)→́
Bool
Alice
??
en
joyed
every
con
fere
nce
7
Qu
an
tifica
tion
with
shift
an
dre
set
We
wan
t:Je
very
con
fere
nceK
=ξs.∀
x.co
nf(x
)⇒
s(x)
:T
hin
gBool
Bool .
Here
the
type
Thin
gBool
Bool
has
the
CP
Stra
nsfo
rm(T
hin
g→
Bool)→
Bool.
Ingen
era
l,αδγ
has
the
CP
Stra
nsfo
rm(α→
γ)→
δ.
Reverse
-en
gin
eer
den
ota
tion
sfo
r“e
very
”an
d“so
me”:
JeveryK
=λr. ξ
s.∀x.r(x
)⇒
s(x):(T
hin
g→
Bool)→́
Thin
gBool
Bool
Jsom
eK=
λr.ξ
s.∃x.r(x
)∧
s(x)
:(T
hin
g→
Bool)→́
Thin
gBool
Bool
Can
now
han
dle
qu
an
tifica
tion
al
nou
nph
rase
sin
an
ypositio
n:
Alice
en
joyed
every
con
fere
nce
[JAliceK
8Jen
joyedK′Je
very
con
fere
nceK
]
=[a
lice8en
joyed
′ξs.∀
x.co
nf(x
)⇒
s(x)]
B[∀
x.co
nf(x
)⇒
(λv.[a
lice8en
joyed
′v])(x
)]
B···
B∀x.co
nf(x
)⇒
alice
8en
joyed
′x
:B
ool
Notio
nof
evalu
atio
n!
Begin
nin
gs
of
psy
cholin
gu
istics.
8
Qu
an
tifier
scop
eam
big
uity
Non
dete
rmin
ismin
natu
ral
lan
gu
age:
Som
estu
den
ten
joyed
every
con
fere
nce
.
∃x.∀
y.en
joyed
(y)(x
)←
linear
scope
∀y.∃
x.en
joyed
(y)(x
)←
inverse
scope
How
togen
era
team
big
uity
?Dete
rmin
isticco
mpositio
n+
Dete
rmin
isticw
ord
mean
ings
Dete
rmin
isticse
nte
nce
mean
ings
Tw
oappro
ach
es:
•N
on
determ
inistic
evalu
atio
nord
er
Qu
an
tifiers
evalu
ate
dearlie
rsco
pe
wid
er.
IH
ierarch
yof
con
trol
opera
tors
Qu
an
tifiers
at
ou
ter
levels
scope
wid
er.
(cf.T
DP
Epaper
at
this
PO
PL
by
Bala
t,D
iC
osm
o,
an
dFio
re)
People
ten
dto
pro
cess
word
sin
the
ord
er
they
are
spoken
.
Pro
nou
ns,
qu
estio
ns,
an
dpola
rityfa
vor
the
seco
nd
appro
ach
,bu
tit
need
ssta
gin
g—
9
Qu
an
tifier
scop
eam
big
uity
with
hie
rarch
y&
stagin
g
[Jsom
estu
den
tK8Je
njo
yedK′Je
very
con
fere
nceK
]
=[(
ξ2s.∃
x.stu
den
t(x)∧
s(x))
8en
joyed
′(
ξ1t.∀
y.co
nf(y
)⇒
t(y))]
0
B[∃
x.stu
den
t(x)∧
(
λv.[v
8en
joyed
′ξ1t.∀
y.co
nf(y
)⇒
t(y)] 2
)(x)]
0
B···
B[∃
x.stu
den
t(x)∧
[x8en
joyed
′ξ1t.∀
y.co
nf(y
)⇒
t(y)] 2
]
0
B[∀
y.co
nf(y
)⇒
(
λv.[∃
x.stu
den
t(x)∧
[x8en
joyed
′v] 2
] 1)(y
)]
0
B[∀
y.co
nf(y
)⇒
[∃x.stu
den
t(x)∧
[x8en
joyed
′y] 2
] 1]
0
Wh
at
is∃
above,
really
?
•Is
ith
igh
er-ord
erabstra
ctsy
nta
x:(T
hin
g→
Bool)→
Bool?
No,
beca
use
then
the
bod
yu
nd
er∃
mu
stbe
pu
re.
•Is
itgen
sym
an
dfi
rst-ord
erabstra
ctsy
nta
x:B
ool→
Bool?
Perh
aps,
bu
tn
eed
toru
leou
tu
nbou
nd
xin
∃x.ξ
f.studen
t(x)
Bstu
den
t(x).
(cf.“S
om
estu
den
ten
joyed
every
con
fere
nce
s/he
org
an
ized
.”)
•Id
eally,
itis
hig
her-o
rder
abstra
ctsy
nta
xsta
ged
ina
lan
gu
age
with
con
trol.1
0
Tw
okin
ds
of
con
trol
hie
rarch
ies
Dan
vy
an
dFilin
ski:
•Post-C
PS
types
look
like
α
(
γ0
δ0
δ1
)
(
γ1
δ2
δ3
)
•N
eed
sgen
sym
for
now
Bark
er
an
dSh
an
:
•Post-C
PS
types
look
like
(
αγ
0γ
1
)
δ0
δ1
•N
od
irect-sty
lete
rms
yet
Both
impro
ve
Hobbs
an
dSh
ieber’s
an
dLew
in’s
qu
an
tifier
scopin
galg
orith
ms:
XD
irectly
com
positio
nal,
not
apost-p
roce
ssing
step
afte
rparsin
g
XSem
an
tically
motiv
ate
dby
delim
ited
con
tinu
atio
ns
•In
tera
ctspro
perly
with
oth
er
lingu
isticsid
eeffe
cts
X(o
ther)
qu
an
tifica
tion
–pro
nou
ns
–qu
estio
ns
11
Ou
tline
XO
verv
iew
XA
simple
gra
mm
atica
lfo
rmalism
XQ
uan
tifica
tion
with
shift
an
dre
set
XQ
uan
tifier
scope
am
big
uity
IPola
rityse
nsitiv
ity
Com
pu
tatio
nal
side
effe
ctsin
the
logica
lm
eta
lan
gu
age
...
...han
dle
s“lin
guistic
side
effects”
Sta
tein
the
logica
lm
eta
lan
gu
age
......h
an
dle
spro
nou
ns
an
dbin
din
g
Con
trol
opera
tors
inth
elo
gica
lm
eta
lan
gu
age
......h
an
dle
squ
an
tifica
tion
an
dpola
ritysen
sitivity
12
Pola
rityse
nsitiv
ity
Th
equ
an
tifiers
“a”,
“som
e”,
an
d“a
nyӠ
all
look
existe
ntia
l:
Did
astu
den
tca
ll?D
idso
me
stud
en
tca
ll?
Did
an
ystu
den
tca
ll?
∃x.stu
den
t(x)∧
called
(x)
Bu
td
on
ot
beh
ave
the
sam
e:
No
stud
en
ten
joyed
som
eco
nfe
ren
ce.
(un
am
big
uou
s∃¬
)N
ostu
den
ten
joyed
aco
nfe
ren
ce.
(am
big
uou
s¬∃,∃¬
)N
ostu
den
ten
joyed
an
yco
nfe
ren
ce.
(un
am
big
uou
s¬∃)
Som
estu
den
ten
joyed
no
con
fere
nce
.(u
nam
big
uou
s∃¬
)
Astu
den
ten
joyed
no
con
fere
nce
.(a
mbig
uou
s¬∃,∃¬
)*A
ny
stud
en
ten
joyed
no
con
fere
nce
.(u
nacce
pta
ble
)
“An
y”
isa
neg
ativ
epola
rityitem
:Very
rou
gh
ly,it
requ
ires
negativ
eco
nte
xts,
such
as
inth
esco
pe
of
“no”.
“Som
e”
isa
positiv
epola
rityitem
:Very
rou
gh
ly,it
isalle
rgic
ton
egativ
eco
nte
xts.
Mean
ing
affe
ctsam
big
uity
an
dacce
pta
bility
!B
ut
linear
ord
er
matte
rsto
o.
13
Ch
ain
ing
an
swer
typ
es
Wh
at
an
un
der-a
ppre
ciate
dfe
atu
reof
shift
an
dre
set!
Γ`
F:(α→́
βγ
2
γ3 )
γ0
γ1
Γ`
E:α
γ1
γ2
→́E
Γ`
F′E
:β
γ0
γ3
Γ`
E:α
γ0
γ1
Γ`
F:(α→̀
βγ
2
γ3 )
γ1
γ2
→̀E
Γ`
E8F
:β
γ0
γ3
···γ
0
γ0
γ1
γ1
γ1
γ2
γ2
γ2
γ3
γ3
···
14
Pola
rityse
nsitiv
ityw
ithan
swer-ty
pe
subty
pin
g
Asta
nd
ard
appro
ach
tom
od
elin
gpola
rityse
nsitiv
ity:
split
the
an
swer
type
Boolin
toa
fam
ilyof
subty
pes.
Bool≤
BoolP
os
Bool≤
BoolN
eg
(inad
ditio
nto
the
usu
al
rule
sfo
rsu
bty
pin
g)
Th
ere
turn
type
of
verb
slik
e“e
njo
yed
”re
main
sB
ool.
Also
,re
strictR
ese
tto
pro
du
ceth
ean
swer
type
BoolorB
oolP
os,
notB
oolN
eg.
Γ`
E:α
βαR
ese
tw
here
β≤
BoolP
os
Γ`
[E]:β
Fin
ally,
refi
ne
the
an
swer
types
for
qu
an
tifiers.
JnoK
:(T
hin
g→
Bool)→́
Thin
gBool
BoolN
eg ,
Jsom
eK:(T
hin
g→
Bool)→́
Thin
gBoolP
os
BoolP
os ,
JaK
:(T
hin
g→
Bool)→́
Thin
gBool
Bool ,
Jan
yK:(T
hin
g→
Bool)→́
Thin
gBoolN
eg
BoolN
eg .
BoolP
os
ε som
e
Bool
no
a
BoolN
egε
an
y
15
Pola
rityse
nsitiv
ity:
revisitin
gem
pirica
ld
ata
Qu
an
tifiers
with
levels
Typ
eR
ead
ing
no
1...so
me
2:B
ool B
oolP
osB
ool
BoolN
eg
BoolP
os···
6→
no
1...so
me
16:
no
2...so
me
1:B
ool B
ool B
oolP
os
BoolP
os
BoolN
eg···
→(∃
¬)
no
1...a
2:B
ool B
ool B
ool
BoolN
eg
Bool···
→(¬∃)
no
1...a
1:B
ool B
ool
Bool
→(¬∃)
no
2...a
1:B
ool B
ool B
ool
Bool
BoolN
eg···
→(∃
¬)
no
1...a
ny
2:B
ool B
oolN
eg
Bool
BoolN
eg
BoolN
eg···
6→
no
1...a
ny
1:B
ool B
ool
BoolN
eg
→(¬∃)
no
2...a
ny
1:B
ool B
ool B
oolN
eg
BoolN
eg
BoolN
eg···
6→
“no
...an
y”
mu
stbe
on
the
sam
ele
vel
“no”
mu
stsco
pe
over
“som
e”
BoolP
os
ε som
e
Bool
no
a
BoolN
egε
an
y
16
Pola
rityse
nsitiv
ity:
revisitin
gem
pirica
ld
ata
Qu
an
tifiers
with
levels
Typ
eR
ead
ing
som
e1...n
o2
:B
ool B
ool B
oolP
os
BoolP
os
BoolN
eg···
→(∃
¬)
som
e1...n
o1
:B
ool B
oolP
os
BoolN
eg
→(∃
¬)
som
e2...n
o1
:B
ool B
oolP
osB
ool
BoolN
eg
BoolP
os···
6→
a1...n
o2
:B
ool B
ool B
ool
Bool
BoolN
eg···
→(∃
¬)
a1...n
o1
:B
ool B
ool
BoolN
eg
→(∃
¬)
a2...n
o1
:B
ool B
ool B
ool
BoolN
eg
Bool···
→(¬∃)
an
y1...n
o2
:B
ool B
ool B
oolN
eg
BoolN
eg
BoolN
eg···
6→
an
y1...n
o1
:B
ool B
oolN
eg
Bool
6→
an
y2...n
o1
:B
ool B
oolN
eg
Bool
BoolN
eg
BoolN
eg···
6→
“no
...an
y”
mu
stbe
on
the
sam
ele
vel,
inth
at
ord
er
“no”
mu
stsco
pe
over
“som
e”
BoolP
os
ε som
e
Bool
no
a
BoolN
egε
an
y
17
Pola
rityse
nsitiv
ity:
revisitin
gem
pirica
ld
ata
Qu
an
tifiers
with
levels
Typ
eR
ead
ing
Every
1stu
den
ten
joyed
som
e1
con
fere
nce
:B
ool B
ool
BoolP
os→
(∀∃)
A1
stud
en
ten
joyed
every
1co
nfe
ren
ce:B
ool B
ool
BoolP
os→
(∃∀)
JeveryK
:(T
hin
g→
Bool)→́
Thin
gBool
BoolP
os
No
1pro
fesso
rgave
every
1stu
den
tso
me
1book
:B
ool B
ool
BoolP
os→
(¬∀∃)
“no
...an
y”
mu
stbe
on
the
sam
ele
vel,
inth
at
ord
er
“no”
mu
stsco
pe
over
“som
e” ,
exce
pt
if“e
very
”in
terv
en
es
BoolP
os
ε som
e
Bool
no
a
every
BoolN
egε
an
y
18
Com
parin
gap
pro
ach
es
tosco
pe
am
big
uity
Tw
oappro
ach
es:
•N
on
dete
rmin
isticevalu
atio
nord
er:
Qu
an
tifiers
evalu
ate
dearlie
rsco
pe
wid
er.
IH
iera
rchy
of
con
trol
opera
tors:
Qu
an
tifiers
at
ou
ter
levels
scope
wid
er.
More
com
ple
xperh
aps,
bu
tca
ptu
res
more
em
pirica
ld
ata
.
Pola
rityite
ms
(an
dpro
nou
ns,
an
dqu
estio
ns)
favor
the
seco
nd
appro
ach
:
No
stud
en
ten
joyed
an
yco
nfe
ren
ce.
(un
am
big
uou
s¬∃)
*A
ny
stud
en
ten
joyed
no
con
fere
nce
.(u
nacce
pta
ble
)
Th
efi
rstse
man
tican
aly
sisof
pola
rityite
ms
toca
ptu
reth
elo
ng-o
bse
rved
sen
sitivity
tolin
ear
ord
er.
Imple
men
ted
ina
substru
ctura
llo
gic
of
delim
ited
con
tinu
atio
ns,
usin
gR
ichard
Moot’s
theore
mpro
ver
Gra
ilfo
rca
tegoria
lgra
mm
ar.
(Join
tw
ork
with
Ch
risB
ark
er.)
19
Su
mm
ary
Con
trol
opera
tors
inth
elo
gica
lm
eta
lan
gu
age
han
dle
squ
an
tifica
tion
an
dpola
rityse
nsitiv
ity.
Ingre
die
nts
of
this
an
aly
sisin
clud
e
•a
con
trol
hie
rarch
y,•
staged
gen
era
tion
of
logica
lfo
rmu
las,
•le
ft-to-rig
ht
evalu
atio
nord
er,
an
d•
chan
gin
gan
dch
ain
ing
an
swer
types.
Th
efi
rstse
man
tican
aly
sisof
pola
rityite
ms
toca
ptu
relin
ear
ord
er.
Beyon
dth
eλ-ca
lculu
s:opera
tion
al
sem
an
ticsfo
rn
atu
ral
lan
gu
age?
20
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